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✓ 12 results · all verified · 2 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 10 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Riemannian Comparison Theorems — Examples

1 · Prerequisites

2 · Summary

The companion page collects the explicit computations that pin the comparison statements of the theory page to concrete numbers and the counterexamples that bound their hypotheses. All items work in the same curvature convention and inherit ${A C}_{ω}$ exactly where the corresponding theory suppliers do.

Model Jacobi fields in the three constant-curvature signs, the Euclidean-versus-spherical Rauch comparison, and the equality of the Hessian, Laplacian, volume and Toponogov comparisons in the model spaces are computed directly from the model functions and the constant-curvature tensor identity. The examples for hyperbolic space and the flat torus test Cartan--Hadamard and the covering-map theorem at both sides: the hyperboloid model is verified complete and globally exponential, while the flat torus shows that simple connectedness cannot be dropped from global injectivity of $\exp_{p}$ ; the Bonnet--Myers sphere example attains the diameter bound with $Ric = (n - 1) k g$ . The two volume examples re-derive the model ball volumes, for all three signs of $k$ and with the exponential growth rate in the hyperbolic case, and the Toponogov example verifies the hinge, triangle and chord equalities on the round sphere as a diagnostic of the direction conventions.

The counterexamples show where the hypotheses are load-bearing: the paraboloid $z = x^{2} + y^{2}$ is complete, noncompact and positively curved with curvature tending to zero, so no fixed positive lower bound may be inferred from pointwise positivity; and the product $H^{2} (- 1) \times R^{n - 2}$ has Ricci curvature bounded below while one sectional curvature stays at $- 1$ , so a Ricci bound controls the trace but not the individual planes. The final diagnostic example records the equality and strict signs of all comparison families in the constant-curvature models and is never used as a dependency of a later proof.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

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Model jacobi fields in positive zero and negative curvature

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $(M^{n}, g)$ , $n \geq 2$ , be a Riemannian manifold of constant sectional curvature $k \in R$ , let $γ : I \to M$ be a unit-speed geodesic on an interval $I \subseteq R$ with $0 \in I$ , write $T = \dot{γ}$ , let $P_{t}$ denote parallel transport along $γ$ , and let $E$ be a vector of the normal space $N_{0} = {X \in T_{γ (0)} M : g (X, T (0)) = 0}$ . Let $J$ be the unique Jacobi field along $γ$ with $J (0) = 0, D_{t} J (0) = E .$ Then, for every $t \in I$ , $J (t) = {sn}_{k} (t) P_{t} E = A (t) E,$ where $A$ is the radial Jacobi tensor. When $E \neq 0$ , the three signs of $k$ give the following separation behaviours:

$k > 0$ : $J (t) = \frac{\sin (\sqrt{k} t)}{\sqrt{k}} P_{t} E$ — the sine branch, whose first positive zero is $t = π / \sqrt{k}$ ; if this time belongs to $I$ , then $J$ vanishes there and the radial field refocuses;
$k = 0$ : $J (t) = t P_{t} E$ — the linear branch, with no zero other than $t = 0$ ;
$k < 0$ : $J (t) = \frac{\sinh (\sqrt{- k} t)}{\sqrt{- k}} P_{t} E$ — the hyperbolic-sine branch, positive and strictly increasing in length for $t > 0$ , with no positive zero.

For $E = 0$ the field is identically zero. All zero-time claims concern only times contained in $I$ .

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1], a real number $k$ , a Riemannian manifold $(M^{n}, g)$ of constant sectional curvature $k$ with $n \geq 2$ , a unit-speed geodesic $γ : I \to M$ with $0 \in I$ , parallel transport $P_{t}$ along $γ$ , a normal vector $E \in N_{0}$ , and the unique Jacobi field $J$ with $J (0) = 0$ , $D_{t} J (0) = E$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), entering through the constant-curvature interface Constant sectional curvature and space form; the Jacobi initial-value construction used in Radial Jacobi tensor requires no choice.

[F1]

Constant curvature: a Riemannian manifold has constant sectional curvature $k$ exactly when $R (X, Y) Z = k (g (Y, Z) X - g (X, Z) Y)$ (Curvature tensor of constant sectional curvature); the predicate and the space-form terminology are those of Constant sectional curvature and space form.

[F2]

The comparison sine: ${sn}_{k}$ is the piecewise function of Comparison sine, cosine and cotangent functions with ${sn}_{k} (0) = 0$ and ${sn}_{k}^{'} (0) = 1$ , positive on $(0, π / \sqrt{k})$ when $k > 0$ , with ${sn}_{k} (π / \sqrt{k}) = 0$ , and with no positive zero when $k \leq 0$ . It satisfies ${sn}_{k}^{''} + k {sn}_{k} = 0$ (Model functions solve the constant curvature jacobi equation).

[F3]

Jacobi fields and radial data: a Jacobi field along $γ$ solves $D_{t}^{2} J + R (J, T) T = 0$ (Jacobi field, Covariant derivative along a curve); for every $w \in N_{0}$ the unique Jacobi field $J_{w}$ with $J_{w} (0) = 0$ and $D_{t} J_{w} (0) = w$ has $A (t) w = J_{w} (t)$ and is normal, $g (J_{w} (t), T (t)) = 0$ (Radial Jacobi tensor).

[F4]

Parallel transport: $P_{t}$ is characterized by $P_{0} = i d$ and $D_{t} (P_{t} E) = 0$ for every $E$ (Parallel section along a curve), and it preserves the metric, hence preserves inner products and normality (Levi civita parallel transport preserves lengths angles and volume).

[F5]

Realizations: the round sphere $S_{R}^{n}$ of Round sphere model geometry is a complete Riemannian manifold of constant sectional curvature $1 / R^{2}$ for every $R > 0$ , and the half-space model of Upper half-space model geometry is a complete Riemannian manifold of constant sectional curvature $- a^{2}$ for every $a > 0$ .

Verification

technique · direct: the curvature of a constant-curvature manifold acts as $k$ times the identity on normal vectors, so the explicit field $\operatorname{sn}_k(t)P_tE$ satisfies the Jacobi equation with the prescribed initial data, and uniqueness identifies it with $A(t)E$; the three signs are then read off the comparison functions

1.1F1F3given

On normal vectors the curvature endomorphism is $k$ times the identity. [F1, F3, given] Let $X$ be a vector field along $γ$ with $X (t) ⊥ T (t)$ for every $t$ . Since $g (T, T) \equiv 1$ , the formula of [F1] gives $R (X, T) T = k (g (T, T) X - g (X, T) T) = k X .$ The computation is pointwise, so it applies at every $t \in I$ and for every normal vector there.

2.1F2F3F4step 1.1

The field $F (t) = {sn}_{k} (t) P_{t} E$ is a Jacobi field with the initial data of $J$ . [F2, F3, F4, step 1.1] Because $D_{t} (P_{t} E) = 0$ by [F4], the covariant derivative along $γ$ acts on the scalar multiple by $D_{t} F = {sn}_{k}^{'} (t) P_{t} E, D_{t}^{2} F = {sn}_{k}^{''} (t) P_{t} E = - k {sn}_{k} (t) P_{t} E,$ where the last equality is the differential equation of [F2]. The parallel field $P_{t} E$ remains normal to $T$ by the metric preservation in [F4], so step 1.1 applies to the field $F$ and gives $R (F, T) T = k F = k {sn}_{k} (t) P_{t} E$ . Adding the two displays, $D_{t}^{2} F + R (F, T) T = 0$ : the field $F$ is a Jacobi field along $γ$ . At $t = 0$ the initial data of [F2] give $F (0) = {sn}_{k} (0) P_{0} E = 0$ and $D_{t} F (0) = {sn}_{k}^{'} (0) E = E$ , because $P_{0}$ is the identity.

3.1F3step 2.1

The model field is the radial field of $E$ . [F3, step 2.1] By [F3] the Jacobi field with $J (0) = 0$ and $D_{t} J (0) = E$ is unique, and $A (t) E$ is by definition that field. Since $F$ of step 2.1 is a Jacobi field with exactly these initial data, $F = J$ , that is $J (t) = A (t) E = {sn}_{k} (t) P_{t} E$ for every $t \in I$ . Both sides are normal fields along $γ$ by [F3] and [F4].

4.1F2F4step 3.1

The three sign branches and their zero sets. [F2, step 3.1] Inserting the piecewise formula for ${sn}_{k}$ from [F2] into step 3.1 gives the three displays of the statement. Since $P_{t}$ preserves lengths, $∣ J (t) ∣ = ∣ {sn}_{k} (t) ∣ ∣ E ∣$ . For $E \neq 0$ , the zeros on $I$ are exactly the multiples of $π / \sqrt{k}$ lying in $I$ when $k > 0$ , and only $t = 0$ when $k \leq 0$ . In particular the first positive spherical zero occurs at $π / \sqrt{k}$ if that time lies in $I$ . For $k < 0$ the scalar profile is positive and strictly increasing on $t > 0$ ; the field grows without bound only if $I$ contains arbitrarily large positive times. For $E = 0$ the field vanishes identically.

5.1F3F5step 1.1step 3.1step 4.1∎

Cases, realizations and consistency. [F3, F5, step 1.1, step 3.1, step 4.1] The case $E = 0$ gives the zero field in step 3.1, and the case $t = 0$ gives $J (0) = 0$ ; the formula is linear in $E$ , so it exhibits $A (t)$ as a linear map $N_{0} \to N_{t}$ as asserted in [F3]. For $k > 0$ the value $k$ is realized by the round sphere $S_{R}^{n}$ of [F5] with $R = 1 / \sqrt{k}$ , and the refocusing time $π / \sqrt{k} = π R$ is exactly the cut time of Round sphere model geometry, so the radial field refocuses at the antipode; for $k < 0$ the half-space model of [F5] with $a = \sqrt{- k}$ realizes the hyperbolic-sine branch. In particular no curvature sign is excluded and no upper bound on the domain $I$ is needed: the identity of step 3.1 holds on all of $I$ , including beyond a zero when $k > 0$ . Only the inherited [A1] choice is used; the parallel fields, the geodesic and the model functions are explicit.

Source locator

Datar §24.1, pp.173–176, computes the model Jacobi fields of the constant curvature spaces and their curvature; Eschenburg §2, pp.6–11, records the three sign branches. The derivation above is carried out locally from the published constant-curvature tensor identity and the in-run comparison-function and radial-tensor items.

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Rauch comparison between euclidean and spherical geodesics

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Compare the radial normal Jacobi fields of the unit round sphere $S^{n}$ (sectional curvature $k = 1$ ) and of Euclidean space $R^{n}$ (curvature $k = 0$ ) with equal unit initial derivatives: along a unit-speed geodesic of $S^{n}$ the spherical field has length $∣ J_{s p h e r e} (t) ∣ = \sin t,$ while along a unit-speed straight line in $R^{n}$ the Euclidean field has length $∣ J_{e u c l i d} (t) ∣ = t .$ Rauch's first comparison gives $\sin t \leq t (0 \leq t \leq π),$ with strict inequality for every interior time $0 < t < π$ . The endpoint $t = π$ is the antipodal conjugate instant of the sphere, where the spherical field returns to zero.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1], the unit round sphere $S^{n}$ and Euclidean space $R^{n}$ , unit-speed geodesics $γ_{1}$ in $S^{n}$ and $γ_{2}$ in $R^{n}$ , and unit normal vectors $E_{i}$ with parallel transports $Φ_{t}^{i} E_{i}$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), used through the Jacobi-field and parallel transport suppliers.

[F1]

Comparison sine (Comparison sine, cosine and cotangent functions, Model functions solve the constant curvature jacobi equation): ${sn}_{1} (t) = \sin t$ and ${sn}_{0} (t) = t$ , ${sn}_{k}^{''} + k {sn}_{k} = 0$ , ${sn}_{k} (0) = 0$ , ${sn}_{k}^{'} (0) = 1$ , and ${sn}_{1} (t) > 0$ for $0 < t < π$ .

[F2]

Model geometry (Constant sectional curvature and space form, Curvature tensor of constant sectional curvature, Jacobi field): the sphere is a manifold of constant curvature $1$ and Euclidean space one of curvature $0$ , so for a parallel normal unit field $Φ_{t} E$ the fields ${sn}_{1} (t) Φ_{t} E$ and ${sn}_{0} (t) Φ_{t} E$ satisfy the Jacobi equation on the respective geodesics.

[F3]

Parallel transport (Existence and uniqueness of parallel sections, Levi civita parallel transport preserves lengths angles and volume): the parallel field with prescribed unit normal value exists, is unique and preserves norms and orthogonality.

[F4]

Rauch comparison, first form (Rauch comparison theorem first form): with the pointwise radial curvature hypothesis and no conjugate point of the first manifold in $(0, T]$ , normal radial fields with equal positive initial-derivative norms satisfy $∣ J_{1} (t) ∣ \leq ∣ J_{2} (t) ∣$ on $[0, T]$ .

[F5]

The strict sine bound (Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3, The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , Sine and cosine defined by their real power series, Parity and the Pythagorean identity for sine and cosine): $\sin 0 = 0$ , $\cos 0 = 1$ , $\cos$ is strictly decreasing on $[0, 2]$ , so $\sin t = \cos ξ t < t$ for $0 < t \leq 2$ and some $ξ \in (0, t)$ ; and $∣ \sin t ∣ \leq 1$ for all $t$ by the Pythagorean identity.

Verification

Proof technique: direct: identify the two explicit model fields, apply Rauch's first form on $[0, T]$ for every $T < π$ , extend to $T = π$ by continuity, and verify strictness in the interior by the mean value theorem.

1.1F1F2F3given

The two explicit fields. [F1, F2, F3, given] Let $γ_{1}$ be a unit-speed great-circle geodesic of $S^{n}$ and $γ_{2}$ a unit-speed straight line in $R^{n}$ ; let $E_{1}, E_{2}$ be unit normal vectors at the starting points and $Φ_{t}^{i} E_{i}$ their parallel transports. Put $J_{s p h e r e} (t) : = {sn}_{1} (t) Φ_{t}^{1} E_{1} = \sin t Φ_{t}^{1} E_{1}, J_{e u c l i d} (t) : = {sn}_{0} (t) Φ_{t}^{2} E_{2} = t Φ_{t}^{2} E_{2} .$ By [F1] and [F2] both are normal Jacobi fields with vanishing value at $0$ and initial-derivative norm $1$ , and by the isometry property of parallel transport [F3], $∣ J_{s p h e r e} (t) ∣ = \sin t, ∣ J_{e u c l i d} (t) ∣ = t .$

2.1F1F2F4step 1.1given

Rauch comparison on $[0, T]$ , $T < π$ . [F1, F2, F4, step 1.1, given] Fix $T < π$ . Every radial sectional curvature of the sphere is $1$ and every radial sectional curvature of Euclidean space is $0$ [F2], so the pointwise curvature hypothesis of [F4] holds with the sphere as the more curved manifold; and the sphere has no conjugate point in $(0, T]$ because the spherical radial fields are $\sin t$ times parallel normal fields, which vanish only at multiples of $π$ [F1]. Applying [F4] to the fields of step 1.1 gives $∣ J_{s p h e r e} (t) ∣ \leq ∣ J_{e u c l i d} (t) ∣, that is, \sin t \leq t (0 \leq t \leq T) .$

3.1F5step 2.1given∎

Extension to the endpoint and strictness. [F5, step 2.1, given] Both sides of $\sin t \leq t$ are continuous on $[0, π]$ , and step 2.1 gives the inequality on $[0, T]$ for every $T < π$ ; hence it holds on $[0, π]$ , where $\sin π = 0 \leq π$ . For strictness let $0 < t < π$ . If $0 < t \leq 2$ , then by [F5] $\sin t = \cos ξ t$ for some $ξ \in (0, t) \subset (0, 2]$ and $\cos ξ < 1$ , so $\sin t < t$ . If $2 < t < π$ , then $\sin t \leq ∣ \sin t ∣ \leq 1 < 2 < t$ by [F5]. In both cases the inequality is strict, while at $t = π$ the spherical field returns to zero together with its antipodal conjugate point. Both model fields are explicit, so the inherited ${A C}_{ω}$ of [A1] is not drawn on beyond its declaration.

Source locator

Datar §25.2–25.3 (printed pp.185–189) uses the sphere-versus-Euclidean and Euclidean-versus-hyperbolic pairs as the basic illustrations of the comparison signs, and §24.1 gives the model fields ${sn}_{k}$ ; Eschenburg §3 (printed p.13) records the same model cases of Rauch I. The verification above is carried out from the in-run Rauch first form and the published trigonometric facts.

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Distance hessian and laplacian in space forms

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $n \geq 2$ and $k \in R$ , and let $(M, g)$ be a complete, connected, boundaryless $n$ -dimensional Riemannian manifold of constant sectional curvature $k$ , that is, a space form of curvature $k$ in the sense of Constant sectional curvature and space form. Let $p \in M$ , let $r : = r_{p} = d_{g} (p, \cdot)$ be the distance from $p$ , and let $q \in M ∖ ({p} \cup Cut (p))$ be a point off $p$ and off the cut locus of $p$ , with $t_{0} : = r (q) > 0$ and $t_{0} < \frac{π}{\sqrt{k}} when k > 0.$ Then:

the Hessian of the distance is the projection onto the normal hyperplane, $Hess r (W, Z) = {ct}_{k} (t_{0}) (g_{q} (W, Z) - d r (W) d r (Z)) for all W, Z \in T_{q} M;$
the Laplace–Beltrami operator of $g$ satisfies $Δ_{g} r (q) = (n - 1) {ct}_{k} (t_{0});$
the space form attains equality in both comparison theorems: the two curvature hypotheses of the Hessian comparison hold with equality on every normal plane, and the Ricci lower bound of the Laplacian comparison holds with equality, so the bounds of Hessian comparison for distance under sectional curvature bounds and Laplacian comparison for distance under a ricci lower bound are attained with equality in the model geometry.

The explicit comparison cotangent values are ${ct}_{0} (t) = \frac{1}{t}, {ct}_{k} (t) = \sqrt{k} \cot (\sqrt{k} t) (k > 0), {ct}_{k} (t) = \sqrt{- k} \coth (\sqrt{- k} t) (k < 0),$ so in particular Euclidean space gives $Hess r = r^{- 1} (g - d r \otimes d r)$ and $Δ_{g} r = (n - 1) / r$ . No choice beyond the inherited ${A C}_{ω}$ is used; the point $q$ and the geodesic to it are supplied by the cited interfaces and no family of directions is selected.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1]; a complete, connected, boundaryless $n$ -dimensional Riemannian manifold $(M, g)$ of constant sectional curvature $k$ with $n \geq 2$ ; a point $p \in M$ ; a point $q \in M ∖ ({p} \cup Cut (p))$ with $t_{0} : = d_{g} (p, q) > 0$ and $t_{0} < π / \sqrt{k}$ when $k > 0$ ; the distance function $r = r_{p}$ ; and the comparison functions ${sn}_{k}$ , ${cs}_{k}$ , ${ct}_{k}$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the exponential, cut-locus, curvature and Riccati interfaces cited below; the Jacobi initial-value construction itself requires no choice; the argument makes no selection of directions, bases or curves from a family.

[F1]

Space forms (Constant sectional curvature and space form): a connected, boundaryless, geodesically complete Riemannian manifold of constant sectional curvature $k$ is a space form of curvature $k$ . By Hopf–Rinow theorem, geodesic completeness of the connected manifold $(M, g)$ is equivalent to metric completeness, and any two points of $M$ are joined by a minimizing geodesic.

[F2]

Constant curvature (Curvature tensor of constant sectional curvature, Sectional curvature, Riemann curvature four-tensor): $(M, g)$ has constant sectional curvature $k$ exactly when $R (X, Y) Z = k (g (Y, Z) X - g (X, Z) Y), Rm (X, Y, Z, W) = k (g (Y, Z) g (X, W) - g (X, Z) g (Y, W)),$ and the sectional curvature of the plane spanned by an orthonormal pair $(u, w)$ is $K (u \land w) = Rm (u, w, w, u)$ ; the four-linear form $Rm$ is linear in each argument.

[F3]

The exponential map off the cut locus (The exponential map is a diffeomorphism on the open tangent cut domain): $D_{p} = {t v : v \in S_{p} M, 0 < t < c_{p} (v)}$ is open in $T_{p} M$ and $\exp_{p} ∣_{D_{p}}$ is a diffeomorphism onto $M ∖ ({p} \cup Cut (p))$ .

[F4]

Gradient of the distance (Gradient of the distance is the outward unit radial field off the base point and the cut locus): if $v \in S_{p} M$ and $0 < t < c_{p} (v)$ , then with $γ (t) = \exp_{p} (t v) = q$ one has $grad r (q) = \dot{γ} (t)$ , the segment $γ ∣_{[0, t]}$ is the minimizing radial geodesic from $p$ to $q$ , and in particular $d_{g} (p, q) = t$ .

[F5]

The gradient represents $d r$ (Riemannian gradient): $g_{x} (grad f (x), W) = d f_{x} (W)$ for every smooth $f$ , every $x \in M$ and every $W \in T_{x} M$ .

[F6]

Hessian of the distance off the cut locus (Hessian of distance in terms of radial jacobi fields, The Hessian of a $C^{2}$ scalar field is symmetric, Distance from p is smooth off p and the cut locus): the distance function $r$ is smooth at $q$ , its Levi-Civita Hessian $\nabla^{2} r$ is a symmetric bilinear form there, and for $T : = \dot{γ} (t_{0}) = grad r (q)$ one has $Hess r (T, Y) = 0$ for every $Y \in T_{q} M$ .

[F7]

Hessian comparison (Hessian comparison for distance under sectional curvature bounds): let $γ$ be the minimizing unit-speed geodesic from $p$ to $q$ with $\dot{γ} (t_{0}) = grad r (q)$ , and put $N : = {\dot{γ} (t_{0})}^{⊥}$ . If $Rm (X, \dot{γ} (s), \dot{γ} (s), X) \geq k ∣ X ∣^{2}$ for all $s \in (0, t_{0}]$ and all $X \in {\dot{γ} (s)}^{⊥}$ , then $Hess r (X, X) \leq {ct}_{k} (t_{0}) g_{q} (X, X)$ for all $X \in N$ ; if the reverse curvature inequality holds, then $Hess r (X, X) \geq {ct}_{k} (t_{0}) g_{q} (X, X)$ there; and $Hess r (grad r, \cdot) = 0$ .

[F8]

Laplacian comparison (Laplacian comparison for distance under a ricci lower bound): if $Ric \geq (n - 1) k g$ , then $Δ_{g} r (q) \leq (n - 1) {ct}_{k} (t_{0})$ .

[F9]

Laplace–Beltrami operator (Laplace–Beltrami operator as the trace of the Hessian): $Δ_{g} f = {tr}_{g} Hess f = \sum_{i = 1}^{n} Hess f (e_{i}, e_{i})$ in a $g$ -orthonormal basis $(e_{1}, \dots, e_{n})$ of the tangent space.

[F10]

Ricci curvature (Ricci curvature, Ricci curvature is symmetric and basis independent): in an orthonormal basis $(e_{1}, \dots, e_{n})$ of $T_{x} M$ , with $Rm (A, B, C, D) = g (R (A, B) C, D)$ , $Ric (X, Y) = \sum_{i = 1}^{n} Rm (e_{i}, X, Y, e_{i})$ .

[F11]

Comparison functions (Comparison sine, cosine and cotangent functions): ${sn}_{k} = \sin (\sqrt{k} \cdot) / \sqrt{k}$ , ${cs}_{k} = \cos (\sqrt{k} \cdot)$ for $k > 0$ , ${sn}_{0} (t) = t$ , ${cs}_{0} = 1$ , and ${sn}_{k} = \sinh (\sqrt{- k} \cdot) / \sqrt{- k}$ , ${cs}_{k} = \cosh (\sqrt{- k} \cdot)$ for $k < 0$ , with ${ct}_{k} = {cs}_{k} / {sn}_{k}$ on the positive domain.

[F12]

Orthonormal bases (Every finite-dimensional real or complex inner product space has an orthonormal basis): every finite-dimensional inner product space has an orthonormal basis.

[F13]

Standard realizations (Round sphere model geometry, The round sphere has positive constant sectional curvature, Euclidean space has zero curvature, $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ , Upper half-space model geometry, Cartan hadamard for hyperbolic space): the round sphere $S_{R}^{n}$ has constant sectional curvature $1 / R^{2}$ , is compact hence geodesically complete, and its cut locus at a point is the antipodal singleton, so $S_{1 / \sqrt{k}}^{n}$ has constant sectional curvature $k$ for $k > 0$ ; Euclidean $R^{n}$ has vanishing curvature tensor and is metrically complete; the upper-half-space metric of the half-space model proposition has constant sectional curvature $- a^{2}$ for every $a > 0$ , that is, $- 1 / r^{2}$ on the scale $r = 1 / a$ , for $n \geq 2$ ; and the hyperboloid model of hyperbolic $n$ -space is an $n$ -dimensional, geodesically and metrically complete manifold of constant sectional curvature $- 1$ .

Proof

technique · direct: since the sectional curvature is identically $k$, both the one-sided curvature hypotheses of the Hessian comparison hold, so the two inequalities collapse to the single normal-Hessian identity; adding the known vanishing on the radial direction gives the full tensor $\operatorname{ct}_k(r)(g-dr\otimes dr)$; tracing it in the orthogonal decomposition $T_qM=N\oplus\mathbb R\operatorname{grad}r$ gives the Laplacian, and tracing the constant-curvature tensor gives the Ricci equality that identifies the space form as an equality case of both comparison theorems

1.1F1F2F3F4F5given

The minimizing geodesic, the normal hyperplane and the curvature data. [F1, F2, F3, F4, F5, given] By [F3] there are a unit vector $v \in S_{p} M$ and a time $t_{1}$ with $0 < t_{1} < c_{p} (v)$ and $\exp_{p} (t_{1} v) = q$ ; let $γ (t) : = \exp_{p} (t v)$ for $t \geq 0$ . By [F4] the segment $γ ∣_{[0, t_{1}]}$ is minimizing from $p$ to $q$ , so $t_{1} = d_{g} (p, q) = t_{0}$ ; hence $q = γ (t_{0})$ with $0 < t_{0} < c_{p} (v)$ and $T : = \dot{γ} (t_{0}) = grad r (q)$ . Put $N : = T^{⊥} \subseteq T_{q} M$ , the orthogonal complement of $T$ . Then $T_{q} M = N \oplus R T$ , and by [F5], $d r (W) = g_{q} (grad r (q), W) = g_{q} (T, W) (W \in T_{q} M),$ so $d r (T) = ∣ T ∣_{g}^{2} = 1$ and $d r$ vanishes on $N$ ; moreover $r (q) = t_{0} > 0$ and $t_{0} < π / \sqrt{k}$ when $k > 0$ by hypothesis, so the domain restriction of [F7] and [F8] is satisfied. Finally, for every $s \in (0, t_{0}]$ and every nonzero $X \in {\dot{γ} (s)}^{⊥}$ the vector $u : = X / ∣ X ∣_{g}$ is a unit vector with $g (u, \dot{γ} (s)) = 0$ , so $(u, \dot{γ} (s))$ is an orthonormal pair and [F2] gives $Rm (X, \dot{γ} (s), \dot{γ} (s), X) = ∣ X ∣_{g}^{2} Rm (u, \dot{γ} (s), \dot{γ} (s), u) = ∣ X ∣_{g}^{2} K (u \land \dot{γ} (s)) = k ∣ X ∣_{g}^{2},$ the last equality because the sectional curvature is constantly $k$ ; the equality holds also for $X = 0$ by four-linearity. Hence both $Rm (X, \dot{γ} (s), \dot{γ} (s), X) \geq k ∣ X ∣_{g}^{2}$ and the reverse inequality hold for every such $X$ , and the geodesic $γ$ is the minimizing unit-speed geodesic from $p$ to $q$ with $\dot{γ} (t_{0}) = grad r (q)$ required by [F7].

2.1F6F7step 1.1

The Hessian on the normal hyperplane. [F6, F7, step 1.1] By step 1.1 both curvature hypotheses of [F7] hold along $γ$ , with $X$ ranging over $N$ and $t_{0} < π / \sqrt{k}$ when $k > 0$ . Applying the first assertion of [F7] (lower curvature bound) and then the second (upper curvature bound) to the same $X \in N$ gives $Hess r (X, X) \leq {ct}_{k} (t_{0}) g_{q} (X, X) \leq Hess r (X, X) .$ The symmetric bilinear forms therefore agree on the diagonal, and polarization gives equality for every $X, Y \in N$ . In addition $Hess r (grad r, \cdot) = 0$ by [F6] (equivalently by the last assertion of [F7]); in particular $Hess r (T, T) = 0$ and, by the symmetry of the Hessian in [F6], $Hess r (T, Y) = Hess r (Y, T) = 0$ for every $Y \in T_{q} M$ .

3.1F5step 1.1step 2.1

The full Hessian tensor. [F5, step 1.1, step 2.1] Let $W, Z \in T_{q} M$ and decompose $W = d r (W) T + X, Z = d r (Z) T + Y,$ where $X : = W - d r (W) T$ and $Y : = Z - d r (Z) T$ lie in $N$ because $d r (X) = d r (W) - d r (W) d r (T) = 0$ and $d r (Y) = 0$ by $d r (T) = 1$ and $d r ∣_{N} = 0$ from step 1.1. Bilinearity of the Hessian, the vanishing of all Hessian entries involving $T$ from step 2.1, and the normal identity of step 2.1 give $Hess r (W, Z) = d r (W) d r (Z) Hess r (T, T) + d r (W) Hess r (T, Y) + d r (Z) Hess r (X, T) + Hess r (X, Y) = {ct}_{k} (t_{0}) g_{q} (X, Y) .$ Since $T ⊥ N$ , orthogonality gives $g_{q} (W, Z) = d r (W) d r (Z) + g_{q} (X, Y)$ , hence $g_{q} (X, Y) = g_{q} (W, Z) - d r (W) d r (Z)$ and $Hess r (W, Z) = {ct}_{k} (t_{0}) (g_{q} (W, Z) - d r (W) d r (Z))$ for all $W, Z \in T_{q} M$ , which is the first displayed conclusion.

4.1F9F12step 3.1

The trace: the Laplacian. [F9, F12, step 3.1] By [F12] the finite-dimensional space $N$ has an orthonormal basis $(e_{1}, \dots, e_{n - 1})$ , and with $e_{n} : = T$ this is an orthonormal basis of $T_{q} M$ because $T_{q} M = N \oplus R T$ , $T ⊥ N$ and $∣ T ∣_{g} = 1$ (step 1.1). By the trace definition [F9] and the Hessian identity of step 3.1, $Δ_{g} r (q) = \sum_{i = 1}^{n - 1} {ct}_{k} (t_{0}) (g_{q} (e_{i}, e_{i}) - d r (e_{i})^{2}) + {ct}_{k} (t_{0}) (g_{q} (T, T) - d r (T)^{2}),$ where $g_{q} (e_{i}, e_{i}) = 1$ and $d r (e_{i}) = 0$ for $i \leq n - 1$ and $g_{q} (T, T) = 1$ , $d r (T) = 1$ . Hence $Δ_{g} r (q) = {ct}_{k} (t_{0}) ((n - 1) - 0) + {ct}_{k} (t_{0}) (1 - 1) = (n - 1) {ct}_{k} (t_{0}),$ which is the second displayed conclusion.

5.1F2F7F8F10step 2.1step 4.1

Equality in both comparison theorems. [F2, F7, F8, F10, step 2.1, step 4.1] The Ricci tensor of a manifold of constant sectional curvature $k$ is $Ric = (n - 1) k g$ : indeed, in an orthonormal basis $(e_{1}, \dots, e_{n})$ of $T_{x} M$ at an arbitrary point $x$ , [F10] and the second display of [F2] give $Ric (X, Y) = \sum_{i = 1}^{n} Rm (e_{i}, X, Y, e_{i}) = \sum_{i = 1}^{n} k (g (X, Y) g (e_{i}, e_{i}) - g (e_{i}, Y) g (X, e_{i})) = k (n g (X, Y) - g (X, Y)),$ that is, $Ric (X, Y) = (n - 1) k g (X, Y)$ . Thus the hypothesis of [F8] holds with equality, and [F8] gives $Δ_{g} r (q) \leq (n - 1) {ct}_{k} (t_{0})$ , which step 4.1 upgrades to equality; so the Laplacian comparison bound is attained. Likewise step 2.1 shows that $Hess r = {ct}_{k} (t_{0}) g$ on $N$ , which is simultaneously the upper bound of the first assertion and the lower bound of the second assertion of [F7]: the model attains equality in both one-sided Hessian comparisons.

6.1F11F13step 3.1step 4.1step 5.1∎

Conclusion, explicit values and the standard realizations. [F11, F13, step 3.1, step 4.1, step 5.1] This proves the three displayed conclusions for every point $q$ off $p$ and off the cut locus with $t_{0} < π / \sqrt{k}$ when $k > 0$ . The explicit values of the comparison cotangent follow from the piecewise formulas of [F11]: ${ct}_{0} (t) = \frac{1}{t}, {ct}_{k} (t) = \frac{\sqrt{k} \cos (\sqrt{k} t)}{\sin (\sqrt{k} t)} = \sqrt{k} \cot (\sqrt{k} t) (k > 0), {ct}_{k} (t) = \sqrt{- k} \coth (\sqrt{- k} t) (k < 0)$ for $t$ in the positive domain, so in Euclidean space the first conclusion reads $Hess r = r^{- 1} (g - d r \otimes d r)$ and the second reads $Δ_{g} r = (n - 1) / r$ , while on the sphere and in hyperbolic space it reads $Δ_{g} r = (n - 1) \sqrt{k} \cot (\sqrt{k} r)$ and $Δ_{g} r = (n - 1) \sqrt{- k} \coth (\sqrt{- k} r)$ . By [F13] the round sphere $S_{1 / \sqrt{k}}^{n}$ for $k > 0$ , Euclidean $R^{n}$ for $k = 0$ , and the constant-curvature models of curvature $k < 0$ built from the hyperbolic half-space metric and the hyperboloid model are realizations of the hypotheses, so the formulas above are their distance Hessians and Laplacians off the pole and the cut locus. The case $n = 2$ is included; the restriction $t_{0} < π / \sqrt{k}$ is vacuous for $k \leq 0$ and excludes the spherical pole for $k > 0$ ; the value $t_{0} = 0$ is excluded because $q \neq p$ . Every object used is supplied by the cited interfaces and no family of directions, bases or curves is selected from a set, so no choice beyond [A1] is used.

Source locator

Datar §§26.1–26.2 and 28.1, pp.191–197 and 205–209, computes the Hessian and Laplacian of the distance function in the constant-curvature model, with the shape operator ${ct}_{k} (r)$ and the saturated comparison at the model pole. Eschenburg §§2–4, pp.6–16, derives the Riccati and Hessian comparison operators whose equality case is the constant-curvature model used here. The two comparison items applied in both directions are the in-run Hessian and Laplacian comparison theorems; the model values are the piecewise comparison functions of Comparison sine, cosine and cotangent functions, and the standard realizations cited in [F13] record the curvature and completeness of the round sphere, Euclidean space and hyperbolic space.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Cartan hadamard for hyperbolic space

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $n \geq 2$ and let $⟨ v, w ⟩ = \sum_{i = 1}^{n} v_{i} w_{i} - v_{n + 1} w_{n + 1}$ be the Lorentz form on $R^{n + 1}$ . The hyperboloid model of hyperbolic $n$ -space is the upper sheet $H^{n} = {x \in R^{n + 1} : ⟨ x, x ⟩ = - 1, x_{n + 1} > 0}$ with the Riemannian metric $g^{H}$ induced by the Lorentz form. Then:

$(H^{n}, g^{H})$ is an $n$ -dimensional Riemannian manifold of constant sectional curvature $K = - 1$ ;
it is geodesically and metrically complete;
it is simply connected;
consequently, for every $p \in H^{n}$ the exponential map $\exp_{p} : (T_{p} H^{n}, \exp_{p}^{*} g^{H}) \to (H^{n}, g^{H})$ is a global diffeomorphism, as Cartan–Hadamard predicts.

The model functions of the comparison page are the hyperbolic-sine branch: the normal Jacobi fields along a unit-speed geodesic of $H^{n}$ , vanishing at parameter $0$ , have the shape ${sn}_{- 1} (t) E (t) = \sinh (t) E (t)$ with $E$ parallel normal, which matches $K = - 1$ and the absence of positive zeros.

Facts & Assumptions

Given: The integer $n \geq 2$ , the Lorentz form $⟨ \cdot, \cdot ⟩$ on $R^{n + 1}$ , the upper sheet $H^{n}$ with its induced metric $g^{H}$ , and the inherited ${A C}_{ω}$ of [A1].

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the Hopf–Rinow, geodesic-existence and Cartan–Hadamard suppliers used below; all manifolds, charts and curves below are explicit.

[F1]

Regular level sets: if $f$ is smooth with $D f (a) \neq 0$ at every point of $f^{- 1} (c)$ , then $f^{- 1} (c)$ is an embedded submanifold of dimension $\dim - 1$ , with $T_{a} (f^{- 1} (c)) = \ker D f (a)$ (A regular level set is an embedded submanifold, The tangent space to a regular level set); an open subset of a smooth manifold carries its canonical restricted smooth structure (An open subset of a smooth manifold has a canonical restricted smooth structure).

[F2]

Pullbacks: the pullback of a smooth covariant tensor field along a smooth map is smooth and functorial (Pullback of covariant tensors is smooth and functorial); a smooth symmetric positive-definite $(0, 2)$ -tensor field is a Riemannian metric (Riemannian metric and riemannian manifold), and the Euclidean Cauchy–Schwarz inequality $∣ u \cdot v ∣ \leq ∣ u ∣ ∣ v ∣$ holds (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F3]

Uniqueness of the Levi-Civita connection: a smooth Riemannian metric has exactly one torsion-free metric-compatible connection (Fundamental theorem of riemannian geometry). For the coordinate directional derivative $D$ on $R^{n + 1}$ one has $D_{X} Y - D_{Y} X = [X, Y]$ in the coordinate Lie bracket (Coordinate formula for the Lie bracket), and $D$ is flat: $D_{X} D_{Y} Z - D_{Y} D_{X} Z - D_{[X, Y]} Z = 0$ for smooth ambient fields, by equality of mixed partial derivatives (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F4]

Geodesics: for every initial datum there is a unique maximal geodesic of a Riemannian manifold, smooth in its arguments (Existence uniqueness and smooth dependence of geodesics), and an affine reparametrization of a geodesic is a geodesic (Affine reparametrization of a geodesic is a geodesic).

[F5]

Hopf–Rinow converts geodesic completeness into metric completeness for a nonempty connected boundaryless Riemannian manifold (Hopf–Rinow theorem); contractible spaces have trivial fundamental group and are simply connected (A contractible space has trivial fundamental group, Simply connected topological spaces); $R^{n}$ is a nonempty convex subset of itself, hence contractible (Every nonempty convex subset of $R^{n}$ is contractible), and path connectedness implies connectedness (Every path-connected space is connected, and every path component lies inside a component).

[F6]

Cartan–Hadamard: a complete, connected, boundaryless Riemannian manifold with $K \leq 0$ that is simply connected has $\exp_{p}$ a diffeomorphism for every $p$ (Cartan hadamard).

[F7]

The model functions: ${sn}_{- 1} (t) = \sinh t$ and ${sn}_{- 1}^{''} - {sn}_{- 1} = 0$ , with ${sn}_{- 1} (0) = 0$ and no positive zero (Model functions solve the constant curvature jacobi equation, Comparison sine, cosine and cotangent functions).

Verification

technique · direct: the hyperboloid is a regular level set; the tangential projection of the flat connection is its Levi-Civita connection; the Gauss expansion gives $K=-1$; explicit hyperbolas are the complete geodesics; the graph projection is a homeomorphism to $\mathbb R^n$

1.1F1given

$H^{n}$ is an embedded $n$ -submanifold of $R^{n + 1}$ and $T_{p} H^{n} = p^{⊥}$ for $p \in H^{n}$ . [F1, given] Put $f (x) = ⟨ x, x ⟩$ , a smooth polynomial function whose differential at $x$ is $D f (x) v = 2 ⟨ x, v ⟩$ . At a point of $f^{- 1} (- 1)$ one has $x \neq 0$ , so $D f (x)$ is not the zero functional and $x$ is a regular point; by [F1] the level set $f^{- 1} (- 1)$ is an embedded submanifold of dimension $n$ with tangent space $\ker D f (x) = x^{⊥}$ . The upper sheet is $H^{n} = f^{- 1} (- 1) \cap {x_{n + 1} > 0}$ , the intersection of that submanifold with an open subset of $R^{n + 1}$ , so [F1] gives it the structure of an embedded $n$ -submanifold with the same tangent spaces.

2.1F2step 1.1

The induced form $g^{H}$ is a Riemannian metric on $H^{n}$ . [F2, step 1.1] Let $ι : H^{n} ↪ R^{n + 1}$ be the inclusion and let $⟨ \cdot, \cdot ⟩$ also denote the ambient covariant two-tensor $h (X, Y) = ⟨ X, Y ⟩$ , which is smooth, symmetric and $C^{\infty}$ -bilinear. Then $g^{H} : = ι^{*} h$ is a smooth symmetric $(0, 2)$ -tensor field by [F2], and it remains to see that it is positive definite on each tangent space. Write $p = (u, q)$ with $u \in R^{n}$ and $q = \sqrt{1 + ∣ u ∣^{2}} > 0$ , and let $v = (v_{0}, v_{n + 1}) \in T_{p} H^{n} = p^{⊥}$ by step 1.1; the orthogonality $⟨ v, p ⟩ = 0$ says $v_{0} \cdot u = v_{n + 1} q$ , that is $v_{n + 1} = (v_{0} \cdot u) / q$ . Hence $g_{p}^{H} (v, v) = ∣ v_{0} ∣^{2} - v_{n + 1}^{2} = ∣ v_{0} ∣^{2} - \frac{(v_{0} \cdot u)^{2}}{1 + ∣ u ∣^{2}} \geq ∣ v_{0} ∣^{2} (1 - \frac{∣ u ∣^{2}}{1 + ∣ u ∣^{2}}) = \frac{∣ v_{0} ∣^{2}}{1 + ∣ u ∣^{2}} \geq 0,$ where [F2] (Cauchy–Schwarz) was used for $(v_{0} \cdot u)^{2} \leq ∣ v_{0} ∣^{2} ∣ u ∣^{2}$ . Equality forces $v_{0} = 0$ and then $v_{n + 1} = 0$ , so $v = 0$ ; therefore $g^{H}$ is positive definite and, by [F2], a Riemannian metric on $H^{n}$ .

2.2F2F5step 1.1

The graph map is a homeomorphism onto $H^{n}$ . [F2, F5, step 1.1] Define $Φ : R^{n} \to H^{n}$ by $Φ (u) = (u, \sqrt{1 + ∣ u ∣^{2}})$ , continuous because the square root is continuous; its image lies in $H^{n}$ , since $∣ u ∣^{2} - (1 + ∣ u ∣^{2}) = - 1$ and the last coordinate is positive. Conversely every $x \in H^{n}$ equals $Φ (x_{1}, \dots, x_{n})$ , because $x_{n + 1}^{2} = 1 + \sum_{i \leq n} x_{i}^{2}$ and $x_{n + 1} > 0$ force $x_{n + 1} = \sqrt{1 + \sum_{i \leq n} x_{i}^{2}}$ . The inverse $Φ^{- 1}$ is the restriction to $H^{n}$ of the continuous projection $x \mapsto (x_{1}, \dots, x_{n})$ , so $Φ$ is a homeomorphism. Therefore $H^{n}$ is path connected (the image of the path-connected $R^{n}$ ) and hence connected by [F5]; being homeomorphic to the contractible space $R^{n}$ , which is contractible by Every nonempty convex subset of $R^{n}$ is contractible applied to the nonempty convex set $R^{n}$ , it is contractible and so simply connected by [F5].

3.1F3step 1.1step 2.1

The tangential projection of the flat connection is the Levi-Civita connection of $g^{H}$ . [F3, step 1.1, step 2.1] Write $D$ for the coordinate directional derivative on $R^{n + 1}$ and $p$ for the position field, so that $D_{X} p = X$ for every smooth ambient field $X$ . Extend tangent fields on $H^{n}$ smoothly to $R^{n + 1}$ (locally, by extending their coordinate expressions). As in step 1.1 the tangent space at $x \in H^{n}$ is $x^{⊥}$ , so the metric orthogonal projection of an ambient vector $w$ onto $T_{x} H^{n} = x^{⊥}$ along the line $R x$ is $w - \frac{⟨ w, x ⟩}{⟨ x, x ⟩} x = w + ⟨ w, x ⟩ x,$ because $⟨ x, x ⟩ = - 1$ . For tangent fields $X, Y$ differentiate the identically vanishing function $⟨ Y, p ⟩$ in the direction $X$ : $0 = X ⟨ Y, p ⟩ = ⟨ D_{X} Y, p ⟩ + ⟨ Y, D_{X} p ⟩ = ⟨ D_{X} Y, p ⟩ + ⟨ X, Y ⟩,$ using $D_{X} p = X$ . Hence the tangential projection of $D_{X} Y$ is $\nabla_{X}^{H} Y : = D_{X} Y + ⟨ D_{X} Y, p ⟩ p = D_{X} Y - ⟨ X, Y ⟩ p,$ which is tangent because $D_{X} Y$ and $p$ are ambient and the correction removes the normal component. The assignment $\nabla^{H}$ is a connection on $H^{n}$ : it is $C^{\infty}$ -linear in $X$ , additive in $Y$ , and $\nabla_{X}^{H} (f Y) = f \nabla_{X}^{H} Y + (X f) Y$ for smooth $f$ , all read off from the same properties of $D$ . It is torsion-free, since for tangent fields $\nabla_{X}^{H} Y - \nabla_{Y}^{H} X = D_{X} Y - D_{Y} X = [X, Y]$ by [F3] (the correction terms cancel because $⟨ X, Y ⟩$ is symmetric), and $[X, Y]$ is tangent to $H^{n}$ because it is a difference of tangential derivatives. It is metric compatible, since $X ⟨ Y, Z ⟩ = ⟨ D_{X} Y, Z ⟩ + ⟨ Y, D_{X} Z ⟩$ and $⟨ \nabla_{X}^{H} Y, Z ⟩ + ⟨ Y, \nabla_{X}^{H} Z ⟩ = ⟨ D_{X} Y, Z ⟩ - ⟨ X, Y ⟩ ⟨ p, Z ⟩ + ⟨ Y, D_{X} Z ⟩ - ⟨ X, Z ⟩ ⟨ Y, p ⟩ = ⟨ D_{X} Y, Z ⟩ + ⟨ Y, D_{X} Z ⟩,$ both correction terms vanishing because $Y, Z ⊥ p$ . By the uniqueness clause of [F3], $\nabla^{H}$ is the Levi-Civita connection of the Riemannian metric $g^{H}$ of step 2.1.

4.1F3step 3.1

The curvature tensor of $g^{H}$ . [F3, step 3.1] For tangent fields $X, Y, Z$ on $H^{n}$ extended as above, use $\nabla_{X}^{H} Y = D_{X} Y - ⟨ X, Y ⟩ p$ and $D_{X} p = X$ to expand $\nabla_{X}^{H} \nabla_{Y}^{H} Z = D_{X} D_{Y} Z - ⟨ Y, Z ⟩ X - (X ⟨ Y, Z ⟩ + ⟨ X, D_{Y} Z ⟩) p,$ because the derivative of the function $⟨ Y, Z ⟩$ is the function $X ⟨ Y, Z ⟩$ and the derivative of $p$ in the direction $X$ is $X$ . Substituting this and the same expression with $X, Y$ interchanged into the definition of the curvature tensor, and adding the term $- \nabla_{[X, Y]}^{H} Z = - D_{[X, Y]} Z + ⟨ [X, Y], Z ⟩ p$ , the $D$ -difference is zero by the flatness of $D$ in [F3], and the $p$ -coefficient is $- X ⟨ Y, Z ⟩ - ⟨ X, D_{Y} Z ⟩ + Y ⟨ X, Z ⟩ + ⟨ Y, D_{X} Z ⟩ + ⟨ [X, Y], Z ⟩ = 0,$ the four scalar terms expanding, by metric compatibility of $⟨ \cdot, \cdot ⟩$ with $D$ , into $- ⟨ [X, Y], Z ⟩$ and therefore cancelling the bracket term $+ ⟨ [X, Y], Z ⟩$ contributed by $- \nabla_{[X, Y]}^{H} Z$ . What remains is the purely tangential identity $R^{H} (X, Y) Z = ⟨ X, Z ⟩ Y - ⟨ Y, Z ⟩ X .$

5.1F3step 4.1

Every sectional curvature of $g^{H}$ equals $- 1$ . [F3, step 4.1] For $p \in H^{n}$ and an orthonormal pair $(X, Y)$ in $T_{p} H^{n} = p^{⊥}$ with respect to $g^{H}$ , step 4.1 gives $Rm (X, Y, Y, X) = ⟨ R^{H} (X, Y) Y, X ⟩ = ⟨ X, Y ⟩ ⟨ Y, X ⟩ - ⟨ Y, Y ⟩ ⟨ X, X ⟩ = 0 - 1 = - 1,$ where the last two equalities use the orthonormality $(g_{p}^{H} (X, X) = 1$ , $g_{p}^{H} (Y, Y) = 1$ , $g_{p}^{H} (X, Y) = 0)$ . Since the Gram determinant in the denominator of the sectional curvature is $1 \cdot 1 - 0^{2} = 1$ , the sectional curvature of every tangent two-plane is $K = - 1$ , so in particular $K \leq 0$ ; in dimension $n \geq 2$ every tangent space carries such a pair.

5.2F3step 3.1step 4.1

The explicit curves are the geodesics, defined for all time. [F3, step 3.1, step 4.1] Fix $p \in H^{n}$ and $v \in T_{p} H^{n}$ with $g_{p}^{H} (v, v) = 1$ , and define $γ : R \to R^{n + 1}$ by $γ (t) = \cosh t p + \sinh t v$ . Then $⟨ γ, γ ⟩ = \cosh^{2} t ⟨ p, p ⟩ + 2 \sinh t \cosh t ⟨ p, v ⟩ + \sinh^{2} t ⟨ v, v ⟩ = - \cosh^{2} t + \sinh^{2} t = - 1$ and $⟨ γ^{'}, γ^{'} ⟩ = \sinh^{2} t (- 1) + \cosh^{2} t (1) = 1$ , so $γ$ is a unit-speed curve in the level set $f^{- 1} (- 1)$ . Its last coordinate is $γ_{n + 1} (t) = \cosh t p_{n + 1} + \sinh t v_{n + 1}$ ; writing $p = (u, q)$ as in step 2.1 one has $∣ v_{n + 1} ∣ \leq ∣ u ∣ < q = p_{n + 1}$ : indeed $v_{n + 1} = (v_{0} \cdot u) / q$ and $g_{p}^{H} (v, v) = 1$ give $v_{n + 1}^{2} \leq ∣ v_{0} ∣^{2} ∣ u ∣^{2} / q^{2} = (1 + v_{n + 1}^{2}) ∣ u ∣^{2} / q^{2}$ , hence $v_{n + 1}^{2} (q^{2} - ∣ u ∣^{2}) \leq ∣ u ∣^{2}$ , that is $v_{n + 1}^{2} \leq ∣ u ∣^{2}$ ; consequently $p_{n + 1} \pm v_{n + 1} \geq q - ∣ u ∣ > 0$ and, since $\cosh t \mp \sinh t > 0$ for every $t$ , the last coordinate $γ_{n + 1} (t) = \frac{1}{2} (e^{t} (p_{n + 1} + v_{n + 1}) + e^{- t} (p_{n + 1} - v_{n + 1}))$ is positive. Hence $γ$ takes values in the upper sheet $H^{n}$ . Its componentwise second derivative is $γ^{''} = \cosh t p + \sinh t v = γ (t)$ , which is $⟨ \cdot, \cdot ⟩$ -orthogonal to $T_{γ (t)} H^{n} = γ (t)^{⊥}$ ; therefore the tangential component of $γ^{''}$ vanishes: $\nabla_{γ^{'}}^{H} γ^{'} = (γ^{''})^{⊤} = 0$ by the projection formula of step 3.1. Thus $γ$ is a geodesic of $(H^{n}, g^{H})$ , defined on all of $R$ , with $γ (0) = p$ and $γ^{'} (0) = v$ . For a general initial vector $w \in T_{p} H^{n}$ , $w = 0$ gives the constant geodesic and $w \neq 0$ is handled by affine reparametrization [F4] of the unit-speed case with $v = w / ∣ w ∣$ .

6.1F3F4F5step 2.2step 5.1step 5.2

Everything is complete and simply connected. [F3, F4, F5, step 2.2, step 5.1, step 5.2] Every initial datum $(p, w) \in T H^{n}$ is the initial datum of the geodesic constructed in step 5.2, which is defined on all of $R$ ; by the uniqueness clause of [F4] the unique maximal geodesic with those initial data has domain $R$ . Hence $(H^{n}, g^{H})$ is geodesically complete, and it is metrically complete by Hopf–Rinow [F5], the manifold being nonempty, connected (step 2.2) and boundaryless. By step 2.2 it is simply connected, and by step 5.1 it has $K = - 1 \leq 0$ .

7.1F6step 6.1

Cartan–Hadamard gives the global diffeomorphism. [F6, step 6.1] The manifold $(H^{n}, g^{H})$ is complete, connected, boundaryless, nonpositively curved and simply connected by step 6.1, so [F6] applies and $\exp_{p} : (T_{p} H^{n}, \exp_{p}^{*} g^{H}) \to (H^{n}, g^{H})$ is a diffeomorphism for every $p \in H^{n}$ — in particular bijective, in accordance with the Cartan–Hadamard prediction.

8.1F7step 7.1∎

The model fields and the boundary cases. [F7, step 7.1] For a unit-speed geodesic, step 4.1 gives $R (J, T) T = - J$ for normal $J$ . Thus its normal Jacobi equation is $D_{t}^{2} J - J = 0$ (Jacobi field). Given $J (0) = 0$ , let $E$ be the unique parallel field with $E (0) = D_{t} J (0)$ (Existence and uniqueness of parallel sections); its initial value is normal by differentiating $g (J, T) = 0$ , so $E$ stays normal. By [F7], $\sinh (t) E (t)$ solves the same equation and has the same initial data. Jacobi uniqueness (Existence and uniqueness of jacobi fields from initial data) gives $J (t) = \sinh (t) E (t)$ . Such a field has no positive zero when it is nonzero, consistent with $K \leq 0$ . For a constant speed $c > 0$ the corresponding formula is $J (t) = \sinh (c t) P_{t} D_{t} J (0) / c$ ; the factor $c^{2}$ in the curvature term explains why the unit-speed qualification is essential. In dimension $n = 1$ the hyperboloid has no tangent two-plane and no sectional curvature to compute, and it is not a Cartan–Hadamard surface; the case $n \geq 2$ is the one stated. The case $v = 0$ of step 5.2 is the constant geodesic; the case $w \neq 0$ is reduced to unit speed by reparametrization; and the two connected components of ${x : ⟨ x, x ⟩ = - 1}$ are separated by the sign of $x_{n + 1}$ , which is why the upper sheet rather than the whole level set is the model. No choice beyond the inherited [A1] is used: the charts, the projection formula and the explicit geodesics are all canonical.

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A flat torus showing simple connectedness is needed for global exp injectivity

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $n \geq 2$ , let $Z^{n} \subseteq R^{n}$ be the integer lattice acting on $R^{n}$ by translations, and let $T^{n} = R^{n} / Z^{n}$ carry the flat torus metric descended from the Euclidean metric, with quotient map $q : R^{n} \to T^{n}$ . Then:

$(T^{n}, g)$ is complete and has constant sectional curvature $K = 0$ ;
under the chart identification of $T_{p} T^{n}$ with $R^{n}$ , and for $p = [x]$ , the exponential map is the translated quotient map $\exp_{p} (v) = [x + v],$ which is not injective;
$T^{n}$ is not simply connected.

Thus $(T^{n}, g)$ satisfies the completeness and nonpositive-curvature hypotheses of Cartan hadamard — indeed $K = 0$ — while failing only its simple connectedness hypothesis, and the conclusion of that theorem fails as well. Simple connectedness cannot be dropped from Cartan–Hadamard.

Facts & Assumptions

Given: The integer $n \geq 2$ , the integer lattice $Z^{n}$ , the quotient $T^{n} = R^{n} / Z^{n}$ with quotient map $q$ , the flat torus metric $g$ of Flat torus model geometry, and the inherited ${A C}_{ω}$ of [A1].

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the existence and uniqueness theory of geodesics used below; all manifolds, charts and curves below are explicit.

[F1]

The flat torus metric: the quotient $T^{n} = R^{n} / Z^{n}$ carries a Riemannian metric $g$ , unique with $q^{*} g = \sum_{i} d x_{i}^{2}$ , whose quotient charts are boxes with integer-translation transitions; $(T^{n}, g)$ is a connected boundaryless smooth $n$ -manifold on which $q$ is a surjective local isometry (Flat torus model geometry).

[F2]

Coordinate calculus: a curve is a geodesic of a coordinate chart exactly when its coordinate acceleration plus the Christoffel term vanishes (Coordinate geodesic equation), the Christoffel symbols of a metric with constant coordinate matrix vanish (Christoffel formula for the levi civita connection), and the curvature components are given by the coordinate formula (Coordinate formula for the curvature tensor) for the Riemann curvature four-tensor of Riemann curvature four-tensor. The sectional curvature normalizes $Rm (X, Y, Y, X)$ by the Gram determinant of an independent pair (Sectional curvature).

[F3]

The completeness and covering theorem: a local isometry $F : (N, \hat{g}) \to (M, g)$ between connected boundaryless Riemannian manifolds with $N$ complete and nonempty has $F$ a covering map and $(M, g)$ complete (A complete local isometry is a covering map); a local isometry is a smooth local diffeomorphism preserving the metric under the differential (Riemannian isometry and local isometry).

[F4]

The exponential map of a flat torus: for the lattice quotient $R^{n} / Z^{n}$ with the descended flat metric, every fibrewise exponential map has domain all of $T_{[x]} T^{n}$ and satisfies $\exp_{[x]} (v) = [x + v]$ (Flat torus model geometry).

[F5]

Covering theory: a covering map is a surjective local homeomorphism whose points have evenly covered neighbourhoods (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings); a path in the base lifts uniquely once its starting point in the total space is fixed (Existence and uniqueness of path lifts through a covering map), and a homotopy with a lift of its initial map lifts uniquely (Existence and uniqueness of homotopy lifts through a covering map).

[F6]

Fundamental group and simple connectedness: based loop classes form $π_{1}$ (Based loops and the fundamental group), the class of the constant loop $c_{x_{0}}$ is its identity element (Loop classes form the group $π_{1} (X, x_{0})$ under concatenation), and a space is simply connected when it is nonempty, path connected and every $π_{1} (X, x_{0})$ has exactly one element (Simply connected topological spaces).

[F7]

Cartan–Hadamard: a complete, connected, boundaryless Riemannian manifold with $K \leq 0$ that is simply connected has $\exp_{p}$ a diffeomorphism for every $p$ (Cartan hadamard).

[F8]

$(R^{n}, d_{2})$ is a complete metric space for $n \geq 1$ ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ ).

Verification

technique · direct: the quotient charts make $q$ a local isometry from the complete Euclidean space, so the completeness/covering theorem makes $q$ a covering map and the torus complete; the constant coordinate metric kills all curvature; straight lines are the geodesics, so $\exp$ is a translated quotient map and is noninjective; a lift of the standard loop through the covering proves that the loop is essential

1.1F1F3F8given

The quotient map $q$ is a local isometry, a covering map, and $(T^{n}, g)$ is complete. [F1, F3, F8, given] In a periodic chart of [F1] the metric matrix is the identity. The local inverse charts of $q$ are the inverses of the restrictions of $q$ to boxes of side lengths less than $1$ , and in those coordinates $q$ is the identity map of an open subset of $R^{n}$ ; hence the differential of $q$ preserves the Euclidean metric of $R^{n}$ and the flat metric of $T^{n}$ at every point. By [F3] this makes $q$ a local isometry from the boundaryless connected Riemannian manifold $(R^{n}, g_{E})$ onto $(T^{n}, g)$ , where $T^{n}$ is connected and boundaryless by [F1]. The Euclidean space is complete by [F8] and nonempty since $n \geq 2$ . The local isometry is therefore a covering map by [F3], and part 3 of [F3] makes $(T^{n}, g)$ complete.

2.1F2step 1.1

The torus has constant sectional curvature $K = 0$ . [F2, step 1.1] Take a periodic chart of [F1] with coordinates $x^{1}, \dots, x^{n}$ , in which the metric matrix is constantly $I_{n}$ . Its first derivatives vanish identically, so every Christoffel symbol vanishes by the formula of [F2]. Substituting the vanishing symbols into the coordinate formula of [F2] gives $R^{ℓ}_{k i j} = 0$ in the chart, that is, the Riemann curvature four-tensor vanishes there, and since the charts cover $T^{n}$ it vanishes on all of $T^{n}$ . Hence $Rm (X, Y, Y, X) = 0$ for every tangent pair, and dividing by the Gram determinant, which is positive for an independent pair by [F2], gives $K = 0$ at every tangent two-plane; in particular $K \leq 0$ .

2.2F2F4step 1.1

Straight lines are the geodesics and the exponential map is $v \mapsto [x + v]$ . [F2, F4, step 1.1] Fix $x, v \in R^{n}$ and let $γ (t) = q (x + t v)$ for $t \in R$ . In a periodic chart containing $q (x + t v)$ the coordinate expression of $γ$ is $t \mapsto x + t v$ up to a constant integer translation, and its coordinate acceleration is identically zero; since the Christoffel symbols of [F1] vanish in these charts, [F2] makes $γ$ a geodesic. Its initial point is $[x]$ and its initial tangent is the vector identified with $v$ , and it is defined on all of $R$ ; by uniqueness of the maximal geodesic with given initial data the world line is complete. Thus every fibrewise exponential map has domain all of $T_{[x]} T^{n}$ and $\exp_{[x]} (v) = γ (1) = [x + v]$ , the formula recorded in [F4].

3.1step 2.2

The exponential map is the quotient map and is not injective. [step 2.2] The chart identification of $T_{[x]} T^{n}$ with $R^{n}$ is linear, so step 2.2 says that $\exp_{[x]}$ is exactly the quotient projection $q$ composed with the translation $v \mapsto x + v$ ; explicitly, $q (w) = [w]$ and $\exp_{[x]} (v) = q (x + v)$ . Thus it is the quotient projection after translation, and is surjective. Let $z = e_{1} \in Z^{n}$ be the first standard basis vector; this is a nonzero lattice vector because $n \geq 2$ . The tangent vectors $0$ and $z$ are distinct, but step 2.2 gives $\exp_{[x]} (0) = [x] = [x + z] = \exp_{[x]} (z) .$ Hence $\exp_{p}$ is not injective for any $p = [x] \in T^{n}$ .

3.2F5F6step 1.1step 2.2

The torus is not simply connected. [F5, F6, step 1.1, step 2.2] Define $γ : [0, 1] \to T^{n}$ by $γ (s) = [s e_{1}]$ , a based loop at $[0]$ . Its lift starting at $0$ is $s \mapsto s e_{1}$ , so its endpoint is $e_{1} \neq 0$ (Existence and uniqueness of path lifts through a covering map in [F5]). Suppose it were homotopic relative endpoints to the constant loop, via $H : [0, 1]^{2} \to T^{n}$ with $H (s, 0) = γ (s)$ , $H (s, 1) = [0]$ , and $H (0, t) = H (1, t) = [0]$ . Lift $H$ with $\tilde{H} (0, 0) = 0$ by [F5]. The lower edge lifts $γ$ , so $\tilde{H} (1, 0) = e_{1}$ . The right edge lifts the constant path starting at $e_{1}$ , hence is constantly $e_{1}$ . The top edge lifts the constant path starting at $\tilde{H} (0, 1) = 0$ , hence is constantly $0$ . Thus $\tilde{H} (1, 1) = e_{1} = 0$ , a contradiction. Therefore $[γ]$ is not the identity in $π_{1} (T^{n}, [0])$ . Since $T^{n}$ is nonempty and path connected, it is not simply connected by [F6].

4.1F3F7step 2.1step 3.1step 3.2∎

Conclusion, and the hypotheses of Cartan–Hadamard. [F3, F7, step 2.1, step 3.1, step 3.2] By step 1.1 the torus is complete, by step 2.1 it has $K = 0 \leq 0$ , and it is a connected boundaryless Riemannian manifold; by step 3.2 it is not simply connected. It therefore satisfies every hypothesis of Cartan hadamard except simple connectedness, and by step 3.1 the conclusion of that theorem — injectivity of $\exp_{p}$ — fails at every point. Hence simple connectedness cannot be omitted from Cartan–Hadamard. The failure is exactly the deck-group phenomenon: for every $p$ the fibre of $\exp_{p}$ over $p$ is the lattice $Z^{n}$ of step 3.1 translations, and the covering $q$ of step 1.1 is nontrivial precisely because the deck translations are nontrivial. Boundary cases: the word "constant curvature $0$ " is two-plane curvature, so it is asserted only for $n \geq 2$ as in the statement; the vector $z = e_{1}$ used in step 3.1 is nonzero exactly because $n \geq 1$ ; the case $v = 0$ of step 2.2 is the constant geodesic through $[x]$ , and $0$ and $z$ are distinct vectors with the same image, so the failure of injectivity is not an artefact of a degenerate vector. No choice beyond the inherited [A1] is used: the lattice, the charts, the loop and the homotopy argument are all explicit.

Source locator

Datar §20.1 and §24.3, pp.147–149 and 178–179, presents the flat torus as the standard complete flat manifold for which global exponential injectivity fails, with simple connectedness the missing Cartan–Hadamard hypothesis; Eschenburg §5, pp.17–19, uses the same model. The chart, curvature, exponential and homotopy-lifting computations are carried out locally above from the published quotient-chart and covering-space suppliers.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Bonnet-Myers for the round sphere

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $k > 0$ , let $R = 1 / \sqrt{k}$ , let $n \geq 2$ , and let $S_{R}^{n} = {x \in R^{n + 1} : ∣ x ∣ = R}$ carry the Riemannian metric induced from Euclidean $R^{n + 1}$ . Then:

$S_{R}^{n}$ has constant sectional curvature $K = 1 / R^{2} = k$ , hence Ricci curvature $Ric = (n - 1) k g$ ;
$S_{R}^{n}$ is complete and has diameter $diam (S_{R}^{n}, g) = π R = \frac{π}{\sqrt{k}};$
it therefore attains the equality case of the Bonnet–Myers bound $diam \leq π / \sqrt{k}$ : both the Ric lower bound $Ric \geq (n - 1) k g$ and the diameter bound are equalities.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1], a real number $k > 0$ , the radius $R = 1 / \sqrt{k}$ , an integer $n \geq 2$ , the round sphere $S_{R}^{n}$ with its induced Riemannian metric $g$ , and the metric diameter $diam (S_{R}^{n}, g) = \sup_{p, q} d_{g} (p, q)$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the sectional-curvature, cut-locus and Hopf–Rinow interfaces cited below; every point and curve below is explicit.

[F1]

The round sphere $S_{R}^{n}$ with the induced metric has constant sectional curvature $1 / R^{2}$ for $n \geq 2$ (The round sphere has positive constant sectional curvature); this is the constant-curvature predicate of Constant sectional curvature and space form.

[F2]

Constant curvature: a manifold of constant sectional curvature $k$ has curvature tensor $R (X, Y) Z = k (g (Y, Z) X - g (X, Z) Y)$ (Curvature tensor of constant sectional curvature); the Ricci tensor is $Ric (X, Y) = tr (Z \mapsto R (Z, X) Y)$ (Ricci curvature) and equals $\sum_{i} Rm (e_{i}, X, Y, e_{i})$ in an orthonormal basis (Ricci curvature is symmetric and basis independent), with $Rm (A, B, C, D) = g (R (A, B) C, D)$ and $K = Rm (u, v, v, u)$ for an orthonormal pair (Riemann curvature four-tensor, Sectional curvature).

[F3]

Cut locus of the round sphere: for every $p \in S_{R}^{n}$ and every unit $v \in T_{p} S_{R}^{n}$ the cut time is $c_{p} (v) = π R$ and the cut locus is the antipodal singleton $Cut (p) = {- p}$ (Round sphere model geometry); the radial geodesics are the great circles of Great circles as round-sphere geodesics, and for $0 < t < c_{p} (v)$ the radial geodesic is minimizing, $d_{g} (p, \exp_{p} (t v)) = t$ , by the definition of the cut time (Cut time in a unit tangent direction).

[F4]

A compact boundaryless Riemannian manifold is geodesically complete, and each connected component is metrically complete (Compact Riemannian manifolds are geodesically complete), and geodesic completeness gives metric completeness by Hopf–Rinow; the Riemannian distance makes a connected Riemannian manifold a metric space (Riemannian distance on a connected manifold).

[F5]

Bonnet–Myers: a nonempty, complete, connected, boundaryless Riemannian manifold of dimension $n \geq 2$ with $Ric \geq (n - 1) k g$ , $k > 0$ , has $diam \leq π / \sqrt{k}$ (Bonnet myers, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

Verification

Proof technique: direct: the round sphere has constant curvature $k$ , its Ricci tensor is traced from the constant-curvature tensor identity, and the diameter is read off from the cut locus, where the antipode lies at distance $π R$ .

1.1F1given

The round sphere has $K = k$ . [F1, given] By [F1] the sectional curvature is everywhere $1 / R^{2}$ , and $R = 1 / \sqrt{k}$ gives $1 / R^{2} = k$ ; in particular $K = k > 0$ and the space is a space form in the sense of [F1].

2.1F2step 1.1

The Ricci curvature is $Ric = (n - 1) k g$ . [F2, step 1.1] Let $p \in S_{R}^{n}$ and let $(e_{1}, \dots, e_{n})$ be an orthonormal basis of $T_{p} S_{R}^{n}$ . By step 1.1 the manifold has constant sectional curvature $k$ , so [F2] gives $R (A, B) C = k (g (B, C) A - g (A, C) B)$ for all tangent vectors. Substituting into the orthonormal-basis formula of [F2], $Ric (X, Y) = \sum_{i = 1}^{n} Rm (e_{i}, X, Y, e_{i}) = \sum_{i = 1}^{n} k (g (X, Y) g (e_{i}, e_{i}) - g (e_{i}, Y) g (X, e_{i})) = k (n g (X, Y) - g (X, Y)) = (n - 1) k g (X, Y) .$ Hence $Ric = (n - 1) k g$ , and in particular the Bonnet–Myers lower bound holds with equality at every point and every tangent vector.

3.1F3F4givenstep 2.1

The sphere is complete and its diameter is $π / \sqrt{k}$ . [F3, F4, given, step 2.1] The sphere $S_{R}^{n}$ is a closed and bounded subset of $R^{n + 1}$ , hence compact. It is connected and boundaryless by the round-sphere geometry of [F3], so [F4] makes it complete. Diameter: fix $p \in S_{R}^{n}$ . Every point $q \in S_{R}^{n}$ either is the antipode $q = - p$ or lies outside the cut locus ${- p}$ of [F3]. In the second case the distance formula of Round sphere model geometry gives $d_{g} (p, q) < π R$ , including $q = p$ with distance zero; in the first case the radial geodesic in the direction of $v$ with $\exp_{p} (π R v) = - p$ has length $π R$ and is minimizing, because the cut time is exactly $π R$ , so $d_{g} (p, - p) = π R$ . Therefore $\sup_{q} d_{g} (p, q) = π R$ , and since $p$ was arbitrary, $diam (S_{R}^{n}, g) = π R = \frac{π}{\sqrt{k}}$ by the definition of the diameter in [F5].

4.1F5step 2.1step 3.1∎

Equality in Bonnet–Myers, and the boundary cases. [F5, step 2.1, step 3.1] By step 2.1 the sphere satisfies $Ric \geq (n - 1) k g$ with equality everywhere, and it is complete, connected, boundaryless and of dimension $n \geq 2$ ; [F5] gives $diam \leq π / \sqrt{k}$ , and step 3.1 gives $diam = π / \sqrt{k}$ : the diameter bound is attained, so the example is an equality case of Bonnet–Myers. The case $n = 2$ is included and is the minimal dimension for which the statement and Bonnet–Myers are formulated; the parameter $k > 0$ is essential, since for $k = 0$ the Euclidean space has unbounded pairwise distances and Ricci curvature $0$ ; its diameter is undefined under Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space. The radius is normalized to $R = 1 / \sqrt{k}$ so that the curvature is exactly $k$ ; for a general radius $r$ the same computation gives $K = 1 / r^{2}$ and diameter $π r$ . No choice beyond the inherited [A1] is used: the sphere, its antipodal point and the great circles are explicit.

Source locator

Datar §27.1 and §28.2, pp.199–200 and 210–212, and Eschenburg §12, pp.59–62, present the round sphere of curvature $k$ as the equality case of Myers' diameter bound. The curvature and cut-locus computations are those of the published round-sphere items cited in [F1]–[F3].

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Bishop gromov ratio is constant in the model space

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $n \geq 2$ and $k \in R$ , and let $(M, g)$ be a complete, connected, simply connected Riemannian $n$ -manifold of constant sectional curvature $k$ ; when $k > 0$ take $(M, g)$ to be the round sphere $S_{1 / \sqrt{k}}^{n}$ , the simply connected space form of positive curvature $k$ . Then for every $p \in M$ and every radius $r > 0$ the open metric ball $B (p, r)$ has the saturated model volume ${vol}_{g} (B (p, r)) = V_{k}^{⋆} (r),$ and consequently the Bishop–Gromov ratio $R_{p} (r) = {vol}_{g} (B (p, r)) / V_{k}^{⋆} (r)$ is equal to $1$ at every positive radius. The ratio is thus constant in $r$ ; for $k > 0$ this constancy continues past the spherical endpoint $r = π / \sqrt{k}$ , where the numerator and the saturated denominator both saturate. The standard realizations are the round sphere for $k > 0$ , Euclidean $n$ -space for $k = 0$ and hyperbolic $n$ -space for $k < 0$ ; for $k \leq 0$ the computation below applies to every complete, connected, simply connected constant-curvature- $k$ manifold, not only to the named realization.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1]; the dimension $n \geq 2$ ; the curvature $k \in R$ ; a complete, connected, simply connected Riemannian $n$ -manifold $(M, g)$ of constant sectional curvature $k$ , equal to the round sphere $S_{1 / \sqrt{k}}^{n}$ when $k > 0$ ; a point $p \in M$ ; a radius $r > 0$ ; the unit sphere $S_{p} M = {v \in T_{p} M : ∣ v ∣_{g_{p}} = 1}$ ; the polar surface measure $σ_{p}$ ; the radial geodesics $γ_{v} (t) = \exp_{p} (t v)$ with cut times $c_{p} (v)$ ; the radial Jacobi tensor $A (t)$ of $γ_{v}$ and the radial volume Jacobian $J_{p} (t, v)$ ; and the model functions ${sn}_{k}$ , $A_{k}$ , $V_{k}$ and $V_{k}^{⋆}$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the cut-time, polar-integration, Cartan–Hadamard and Hopf–Rinow interfaces below; no further selection is made.

[F1]

Polar integration (Polar integration may discard the cut locus): $(M, g)$ is complete, connected and boundaryless, $σ_{p}$ is the finite Borel measure on $S_{p} M$ obtained by transporting the polar surface measure of the unit sphere, and for every Borel $f : M \to [0, \infty]$ , $\int_{M} f d {vol}_{g} = \int_{S_{p} M} \int_{0}^{c_{p} (v)} f (γ_{v} (t)) \det a_{v} (t) d t d σ_{p} (v),$ where $a_{v} (t)$ is the matrix of the radial Jacobi fields in a parallel orthonormal frame, with $\det a_{v} (t) > 0$ for $0 < t < c_{p} (v)$ .

[F2]

Radial Jacobi tensor and radial volume Jacobian (Radial Jacobi tensor, Radial volume jacobian): for $w$ in the normal space $N_{0} = {X \in T_{p} M : g_{p} (X, v) = 0}$ the map $t \mapsto A (t) w$ is the unique Jacobi field along $γ_{v}$ with $A (0) w = 0$ and $D_{t} (A w) (0) = w$ ; in the parallel identification ${\overset{ˉ}{A}}_{v} (t) = P_{t}^{- 1} \circ A (t)$ one has $J_{p} (t, v) = \det {\overset{ˉ}{A}}_{v} (t) > 0$ for $0 < t < c_{p} (v)$ , and $\det a_{v} (t) = J_{p} (t, v)$ in the notation of [F1]. Moreover $\dim N_{0} = n - 1$ .

[F3]

Model fields and comparison functions (Model jacobi fields in positive zero and negative curvature, Model functions solve the constant curvature jacobi equation, Comparison sine, cosine and cotangent functions, Constant sectional curvature and space form): on a Riemannian manifold of constant sectional curvature $k$ , the Jacobi field with $J (0) = 0$ , $D_{t} J (0) = E \in N_{0}$ satisfies $J (t) = {sn}_{k} (t) P_{t} E$ for every $t$ , so the radial Jacobi tensor is $A (t) = {sn}_{k} (t) P_{t}$ and its parallel-frame matrix is ${\overset{ˉ}{A}}_{v} (t) = {sn}_{k} (t) \cdot {i d}_{N_{0}}$ . The comparison sine is given by ${sn}_{0} (t) = t$ , ${sn}_{- a^{2}} (t) = \sinh (a t) / a$ for $a > 0$ and ${sn}_{k} (t) = \sin (\sqrt{k} t) / \sqrt{k}$ for $k > 0$ , it is positive on its positive domain $(0, π / \sqrt{k})$ for $k > 0$ and $(0, \infty)$ for $k \leq 0$ , and it satisfies ${sn}_{k}^{''} + k {sn}_{k} = 0$ .

[F4]

Cut time and minimizing initial intervals (Cut time in a unit tangent direction, Minimizing along a geodesic is an initial interval property): $c_{p} (v) = \sup {t > 0 : d_{g} (p, γ_{v} (t)) = t} \in (0, + \infty]$ , the set $A_{p} (v) = {t \geq 0 : d_{g} (p, γ_{v} (t)) = t}$ is an initial interval, and if $c_{p} (v)$ is finite then $c_{p} (v) \in A_{p} (v)$ . Hence $d_{g} (p, γ_{v} (t)) = t$ for every $0 < t < c_{p} (v)$ : if $t < c_{p} (v) = \sup A_{p} (v)$ there is $t^{'} \in A_{p} (v)$ with $t^{'} > t$ , and the initial-interval property gives $t \in A_{p} (v)$ .

[F5]

Cartan–Hadamard (Cartan hadamard): a connected, boundaryless, finite-dimensional Riemannian manifold that is complete and has sectional curvature $K \leq 0$ everywhere has $\exp_{p} : T_{p} M \to M$ a smooth covering map, and if it is in addition simply connected then $\exp_{p}$ is a diffeomorphism for every $p$ ; in particular $\exp_{p}$ is then injective.

[F6]

Hopf–Rinow (Hopf–Rinow theorem): for a nonempty, connected, boundaryless Riemannian manifold, metric completeness, geodesic completeness and $E_{p} = T_{p} M$ are equivalent, and then every $x, y \in M$ are joined by a minimizing geodesic: there is $w \in T_{x} M$ with $\exp_{x} (w) = y$ and $∣ w ∣_{g_{x}} = d_{g} (x, y)$ .

[F7]

The round sphere (The round sphere has positive constant sectional curvature, Round sphere model geometry): for $k > 0$ and $R = 1 / \sqrt{k}$ , the round sphere $S_{R}^{n}$ with its induced metric is complete, has constant sectional curvature $1 / R^{2} = k$ , and for every $p \in S_{R}^{n}$ and every unit $v \in T_{p} S_{R}^{n}$ the cut time is $c_{p} (v) = π R = π / \sqrt{k}$ .

[F8]

Model volumes and the sphere measure (Model space radial area and ball volume, The polar surface set function on the unit sphere): with $ω_{n - 1}$ the total surface measure of the unit sphere $S^{n - 1} \subseteq R^{n}$ , the model radial area is $A_{k} (s) = ω_{n - 1} {sn}_{k} (s)^{n - 1}$ , the model ball volume is $V_{k} (r) = \int_{0}^{r} A_{k} (t) d t$ , and the saturated model volume is $V_{k}^{⋆} (r) = V_{k} (\min {r, π / \sqrt{k}})$ for $k > 0$ and $V_{k}^{⋆} (r) = V_{k} (r)$ for $k \leq 0$ . The measure $σ_{p}$ of [F1] is the transport of the polar surface measure of $S^{n - 1}$ by a linear isometry, hence $σ_{p} (S_{p} M) = ω_{n - 1}$ , the total surface measure being preserved by the bijection.

[F9]

Borel indicator (The Borel sigma-algebra of a topological space): the open ball $B (p, r)$ is an open subset of $M$ , hence a Borel set, and the indicator of a Borel set is a Borel function: the preimage of an open subset of $R$ under $1_{B (p, r)}$ is one of $\emptyset$ , $B (p, r)$ , $M ∖ B (p, r)$ or $M$ , according to whether the open set contains neither, only $1$ , only $0$ , or both of the values $0, 1$ . All four sets are Borel.

[F10]

Ricci curvature of a space form of curvature $k$ (Distance hessian and laplacian in space forms): such a manifold satisfies $Ric = (n - 1) k g$ ; in particular the Ricci lower bound $Ric \geq (n - 1) k g$ holds with equality.

[F11]

Bishop–Gromov comparison (Bishop gromov volume comparison): for a complete, connected, boundaryless $(M, g)$ of dimension $n \geq 2$ with $Ric \geq (n - 1) k g$ , the ratio $R_{p} (r) = {vol}_{g} (B (p, r)) / V_{k}^{⋆} (r)$ is nonincreasing on $(0, \infty)$ with $\lim_{r ↓ 0} R_{p} (r) = 1$ , it is constant on $[π / \sqrt{k}, \infty)$ when $k > 0$ , and ${vol}_{g} (B (p, r)) \leq V_{k}^{⋆} (r)$ for every $r > 0$ .

Verification

Proof technique: direct: the radial Jacobi tensor of a constant-curvature $k$ manifold is ${sn}_{k} (t)$ times parallel transport, so the polar integration formula writes the ball volume as $ω_{n - 1}$ times the integral of the model density cut off at the cut time; the cut time is $+ \infty$ for $k \leq 0$ by Cartan–Hadamard and $π / \sqrt{k}$ for the round sphere, and the saturated model volume reproduces exactly this cutoff.

1.1F2F3given

The radial density of the model. [F2, F3, given] Fix $v \in S_{p} M$ . By [F3], for every $w \in N_{0}$ the radial field is $A (t) w = {sn}_{k} (t) P_{t} w$ , so the parallel-frame matrix of $A (t)$ is the scalar matrix ${\overset{ˉ}{A}}_{v} (t) = {sn}_{k} (t) \cdot {i d}_{N_{0}}$ . Since $\dim N_{0} = n - 1$ by [F2], its determinant is $\det {\overset{ˉ}{A}}_{v} (t) = {sn}_{k} (t)^{n - 1}, 0 < t < c_{p} (v),$ and [F2] identifies this determinant with the radial volume Jacobian, $\det a_{v} (t) = J_{p} (t, v) = {sn}_{k} (t)^{n - 1}$ , in the notation of [F1].

1.2F4F5F6F7given

The cut time of the model in each curvature regime. [F4, F5, F6, F7, given] Let $v \in S_{p} M$ . Suppose first that $k \leq 0$ . Then $(M, g)$ is complete, connected, boundaryless and simply connected with constant sectional curvature $k \leq 0$ , so [F5] makes $\exp_{p} : T_{p} M \to M$ a diffeomorphism, in particular injective. Let $t > 0$ and $q : = γ_{v} (t) = \exp_{p} (t v)$ . By [F6] there is $w \in T_{p} M$ with $\exp_{p} (w) = q$ and $∣ w ∣_{g_{p}} = d_{g} (p, q)$ . Injectivity gives $w = t v$ , so $d_{g} (p, q) = ∣ w ∣_{g_{p}} = t$ ; since $t > 0$ was arbitrary, the set $A_{p} (v)$ of [F4] contains every positive time, hence $c_{p} (v) = + \infty$ . Suppose now that $k > 0$ , so that $(M, g) = S_{1 / \sqrt{k}}^{n}$ by the hypothesis of the example; then [F7] gives $c_{p} (v) = π / \sqrt{k}$ for every unit $v$ . Consequently $\min {c_{p} (v), r} = r$ when $k \leq 0$ and $\min {c_{p} (v), r} = \min {r, π / \sqrt{k}}$ when $k > 0$ , for every $r > 0$ .

2.1F1F4F8F9step 1.1step 1.2given

The ball volume as the model integral. [F1, F4, F8, F9, step 1.1, step 1.2, given] Fix the point $p$ and the radius $r > 0$ . By [F9] the ball $B (p, r)$ is Borel and $1_{B (p, r)}$ is a Borel function, so the polar formula [F1] applies to $f = 1_{B (p, r)}$ : ${vol}_{g} (B (p, r)) = \int_{S_{p} M} \int_{0}^{c_{p} (v)} 1_{B (p, r)} (γ_{v} (t)) \det a_{v} (t) d t d σ_{p} (v) .$ For $0 < t < c_{p} (v)$ the minimizing property of [F4] gives $d_{g} (p, γ_{v} (t)) = t$ , hence $1_{B (p, r)} (γ_{v} (t)) = 1_{{t < r}}$ , while step 1.1 gives $\det a_{v} (t) = {sn}_{k} (t)^{n - 1}$ . Therefore the inner integral equals the extended nonnegative integral $\int_{0}^{\min {c_{p} (v), r}} {sn}_{k} (t)^{n - 1} d t,$ a finite real number. By step 1.2 this number depends on $v$ only through the case distinction: it equals $\int_{0}^{r} {sn}_{k} (t)^{n - 1} d t$ when $k \leq 0$ and $\int_{0}^{\min {r, π / \sqrt{k}}} {sn}_{k} (t)^{n - 1} d t$ when $k > 0$ . Write $I (r)$ for this common value. The outer integrand of the polar formula is then the constant $I (r)$ , and $σ_{p}$ is a finite measure with $σ_{p} (S_{p} M) = ω_{n - 1}$ by [F8], so ${vol}_{g} (B (p, r)) = ω_{n - 1} I (r) .$

3.1F8step 1.2step 2.1given

The volume is the saturated model volume. [F8, step 1.2, step 2.1, given] Put $m : = {\begin{cases} r, & k \leq 0, \\ \min {r, π / \sqrt{k}}, & k > 0. \end{cases}$ By step 2.1 the ball volume is $ω_{n - 1} I (r)$ , and by step 1.2 the inner integral $I (r)$ is exactly $\int_{0}^{m} {sn}_{k} (t)^{n - 1} d t$ in both cases. Since $A_{k} (t) = ω_{n - 1} {sn}_{k} (t)^{n - 1}$ by [F8], ${vol}_{g} (B (p, r)) = \int_{0}^{m} A_{k} (t) d t = V_{k} (m) = V_{k}^{⋆} (r),$ the last equality being the case distinction defining the saturated model volume in [F8]. Moreover $V_{k}^{⋆} (r) > 0$ : the integrand $A_{k}$ is continuous and positive on the nondegenerate interval $(0, m)$ , by the positivity of ${sn}_{k}$ on its positive domain. The value $r = π / \sqrt{k}$ for $k > 0$ is included, since the definition of $V_{k}^{⋆}$ cuts off at that endpoint.

4.1F8F10F11step 1.2step 3.1given∎

Unit ratio, saturation and sharpness of the comparison. [F8, F10, F11, step 1.2, step 3.1, given] By step 3.1, ${vol}_{g} (B (p, r)) = V_{k}^{⋆} (r) > 0$ for every $p \in M$ and every $r > 0$ , so the Bishop–Gromov ratio is $R_{p} (r) = 1$ at every positive radius: it is a constant function of $r$ . When $k > 0$ and $r \geq π / \sqrt{k}$ , step 1.2 makes the cutoff $m = π / \sqrt{k}$ independent of $r$ , so both ${vol}_{g} (B (p, r))$ and $V_{k}^{⋆} (r) = V_{k} (π / \sqrt{k})$ are constant there; this is the saturation clause of [F8] and of the comparison [F11]. Finally, by [F10] the model space satisfies $Ric = (n - 1) k g$ , so the hypotheses of the Bishop–Gromov comparison [F11] hold, and its general conclusion $R_{p} (r) \leq 1$ with limit $1$ at the origin is attained with equality at every radius: the model space is an equality case of the comparison, and the comparison is sharp. The cases $n = 2$ , where $A_{k} = 2 π {sn}_{k}$ , and $r$ below, equal to, or above $π / \sqrt{k}$ (for $k > 0$ ) are all covered by the single cutoff $m$ ; the value $r = 0$ is excluded, as in the definition of $R_{p}$ . No choice beyond the inherited [A1] is used: the point, the radial direction $v$ , the minimizing geodesic of [F6] and the sphere realization of [F7] are fixed or explicit, and the polar, cut-time, Jacobi, Cartan–Hadamard and Hopf–Rinow interfaces carry exactly ${A C}_{ω}$ .

Source locator

Datar §§27.2 and 28.1, pp.200–209, computes the polar volume element with the model radial density ${sn}_{k}^{n - 1}$ and the model ball volume that the Bishop–Gromov quotient compares with; Eschenburg §§4–5, pp.15–20, introduces the model radial density and its integral. The computation above is carried out on the model space itself: the radial Jacobi tensor is ${sn}_{k} (t)$ times parallel transport, the cut time is $+ \infty$ in nonpositive curvature by Cartan–Hadamard and $π / \sqrt{k}$ on the round sphere, and the polar formula turns the ball volume into the saturated model volume $V_{k}^{⋆} (r)$ .

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Volume growth in euclidean and hyperbolic space

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $n \geq 2$ and $a > 0$ , and let $(M, g)$ be a complete, connected, simply connected Riemannian $n$ -manifold of constant sectional curvature $k \leq 0$ . We compute the two cases $k = 0$ and $k = - a^{2}$ , the flat model and the hyperbolic model. Then for every $p \in M$ and every radius $r > 0$ :

Flat case. If $k = 0$ then ${vol}_{g} (B (p, r)) = V_{0} (r) = ω_{n - 1} r^{n} / n$ ; the standard realization is Euclidean $n$ -space, and $V_{0}$ grows polynomially.
Hyperbolic case. If $k = - a^{2}$ then ${vol}_{g} (B (p, r)) = V_{- a^{2}} (r) = ω_{n - 1} \int_{0}^{r} {(\frac{\sinh (a t)}{a})}^{n - 1} d t,$ and for every $r \geq 2 / a$ the volume obeys the explicit lower bound $V_{- a^{2}} (r) \geq \frac{ω_{n - 1} r}{2 (4 a)^{n - 1}} e^{a (n - 1) r / 2},$ so it grows at least exponentially in $r$ at rate $a (n - 1) / 2 > 0$ ; the standard realization is hyperbolic $n$ -space of curvature $- a^{2}$ . The exponential rate degenerates exactly when $n = 1$ , which is why the statement is made for $n \geq 2$ .

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1]; the dimension $n \geq 2$ ; a real number $a > 0$ ; a complete, connected, simply connected Riemannian $n$ -manifold $(M, g)$ of constant sectional curvature $k \leq 0$ ; a point $p \in M$ ; a radius $r > 0$ ; the unit sphere $S_{p} M$ , the polar surface measure $σ_{p}$ , the radial geodesics $γ_{v} (t) = \exp_{p} (t v)$ with cut times $c_{p} (v)$ , the radial Jacobi tensor $A (t)$ and radial volume Jacobian $J_{p} (t, v)$ ; and the model functions ${sn}_{k}$ , $A_{k}$ , $V_{k}$ and $V_{k}^{⋆}$ .

[A1]

[F1]

Comparison functions (Comparison sine, cosine and cotangent functions, Model functions solve the constant curvature jacobi equation): ${sn}_{0} (t) = t$ , and for $k < 0$ , ${sn}_{k} (t) = \sinh (\sqrt{- k} t) / \sqrt{- k}$ ; in particular ${sn}_{- a^{2}} (t) = \sinh (a t) / a$ because $\sqrt{a^{2}} = a$ . Moreover ${sn}_{k} (0) = 0$ , ${sn}_{k}^{'} (0) = 1$ , and ${sn}_{k} (t) > 0$ for every $t > 0$ when $k \leq 0$ .

[F2]

Model volumes (Model space radial area and ball volume): the model radial area is $A_{k} (t) = ω_{n - 1} {sn}_{k} (t)^{n - 1}$ on the positive domain of ${sn}_{k}$ , which is all of $(0, \infty)$ for $k \leq 0$ ; the model ball volume is $V_{k} (r) = \int_{0}^{r} A_{k} (t) d t$ , and for $k \leq 0$ the saturated model volume is $V_{k}^{⋆} (r) = V_{k} (r)$ . Here $ω_{n - 1}$ is the total surface measure of the unit sphere $S^{n - 1} \subseteq R^{n}$ .

[F3]

Radial density in constant curvature (Radial Jacobi tensor, Radial volume jacobian, Model jacobi fields in positive zero and negative curvature): on a manifold of constant sectional curvature $k$ the radial Jacobi tensor of a unit-speed geodesic is $A (t) = {sn}_{k} (t) P_{t}$ , its parallel-frame matrix is the scalar matrix ${sn}_{k} (t) \cdot {i d}_{N_{0}}$ with $\dim N_{0} = n - 1$ , and the radial volume Jacobian equals $\det a_{v} (t) = J_{p} (t, v) = {sn}_{k} (t)^{n - 1}, 0 < t < c_{p} (v) .$

[F4]

Polar integration (Polar integration may discard the cut locus, The polar surface set function on the unit sphere): $(M, g)$ is complete, connected and boundaryless; $σ_{p}$ is the finite Borel measure on $S_{p} M$ obtained by transporting the polar surface measure of the unit sphere by a linear isometry, so $σ_{p} (S_{p} M) = ω_{n - 1}$ by [F2]; and for every Borel $f : M \to [0, \infty]$ , $\int_{M} f d {vol}_{g} = \int_{S_{p} M} \int_{0}^{c_{p} (v)} f (γ_{v} (t)) \det a_{v} (t) d t d σ_{p} (v) .$

[F5]

Cut time and minimizing initial intervals (Cut time in a unit tangent direction, Minimizing along a geodesic is an initial interval property): $c_{p} (v) = \sup {t > 0 : d_{g} (p, γ_{v} (t)) = t} \in (0, + \infty]$ , the minimizing-time set $A_{p} (v)$ is an initial interval, and $d_{g} (p, γ_{v} (t)) = t$ for every $0 < t < c_{p} (v)$ .

[F6]

Cartan–Hadamard and Hopf–Rinow (Cartan hadamard, Hopf–Rinow theorem): a complete, connected, boundaryless manifold with $K \leq 0$ and simply connected has $\exp_{p}$ a diffeomorphism for every $p$ ; Hopf–Rinow says that on a metrically complete connected manifold every $x, y$ are joined by a minimizing geodesic: there is $w \in T_{x} M$ with $\exp_{x} (w) = y$ and $∣ w ∣_{g_{x}} = d_{g} (x, y)$ .

[F7]

Borel indicator (The Borel sigma-algebra of a topological space): the open ball $B (p, r)$ is a Borel set and $1_{B (p, r)}$ is a Borel function, its preimages of open subsets of $R$ being one of $\emptyset$ , $B (p, r)$ , $M ∖ B (p, r)$ or $M$ , according to whether the open set contains neither, only $1$ , only $0$ , or both of the values $0, 1$ . All four sets are Borel.

[F8]

Power integral (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion): for every integer $m \geq 0$ and every $0 \leq s \leq r$ , $\int_{s}^{r} t^{m} d t = \frac{r^{m + 1} - s^{m + 1}}{m + 1};$ indeed $t \mapsto t^{m + 1} / (m + 1)$ has derivative $t^{m}$ by the power rule, and $t^{m}$ is continuous hence integrable on $[s, r]$ , so the second fundamental theorem applies.

[F9]

Hyperbolic and exponential data (The six hyperbolic functions and their natural domains, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential function is strictly increasing, The elementary numerical bound $2 < e < 3$ ): $\sinh x = (e^{x} - e^{- x}) / 2$ with $e^{- x} = 1 / e^{x} > 0$ ; the exponential function is strictly increasing and $e > 2$ . Hence for $x \geq 1$ one has $e^{- x} \leq e^{0} = 1$ and $e^{x} \geq e > 2$ , so $\sinh x \geq (e^{x} - 1) / 2 \geq (e^{x} - e^{x} / 2) / 2 = e^{x} / 4$ .

[F10]

Integral estimates (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ ): for integrable $f \leq g$ on $[s, t]$ one has $\int_{s}^{t} f \leq \int_{s}^{t} g$ , and for $s < c < t$ , $\int_{s}^{t} f = \int_{s}^{c} f + \int_{c}^{t} f$ ; in particular a nonnegative integrand satisfies $\int_{s}^{t} f \geq \int_{c}^{t} f$ .

[F11]

Realizations (Euclidean space has zero curvature, $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ , Upper half-space model geometry): Euclidean $R^{n}$ has identically zero Riemann curvature and is metrically complete by the Euclidean-completeness theorem, so it is the flat case $k = 0$ ; the upper-half-space metric of the half-space model proposition with parameter $a > 0$ has constant sectional curvature $- a^{2}$ for $n \geq 2$ , realizing the hyperbolic curvature normalization.

[F12]

Bishop–Gromov and Ricci of space forms (Bishop gromov volume comparison, Distance hessian and laplacian in space forms): a Riemannian manifold of constant sectional curvature $k$ has $Ric = (n - 1) k g$ ; and a complete, connected, boundaryless $n$ -manifold with $Ric \geq (n - 1) k g$ has ratio $R_{p} (r) = {vol}_{g} (B (p, r)) / V_{k}^{⋆} (r)$ nonincreasing on $(0, \infty)$ with limit $1$ as $r ↓ 0$ and ${vol}_{g} (B (p, r)) \leq V_{k}^{⋆} (r)$ for every $r > 0$ .

Verification

Proof technique: direct: in nonpositive constant curvature the radial Jacobi tensor is ${sn}_{k} (t)$ times parallel transport and Cartan–Hadamard removes all cut points, so the polar formula integrates the model density to the model volume; inserting ${sn}_{0} (t) = t$ and ${sn}_{- a^{2}} (t) = \sinh (a t) / a$ gives the two closed forms, and the lower bound $\sinh x \geq e^{x} / 4$ for $x \geq 1$ converts the hyperbolic integral into an exponential lower bound.

1.1F3given

The radial density of the nonpositively curved model. [F3, given] Let $v \in S_{p} M$ and let $γ_{v} (t) = \exp_{p} (t v)$ be the radial geodesic, a unit-speed geodesic along which the sectional curvature is constantly $k \leq 0$ . By [F3] the radial Jacobi tensor is $A (t) = {sn}_{k} (t) P_{t}$ and the radial volume Jacobian is $\det a_{v} (t) = J_{p} (t, v) = {sn}_{k} (t)^{n - 1}, 0 < t < c_{p} (v) .$

1.2F5F6given

There are no cut points in nonpositive curvature. [F5, F6, given] The manifold is complete, connected, boundaryless and simply connected with sectional curvature $K = k \leq 0$ , so by [F6] the exponential map $\exp_{p} : T_{p} M \to M$ is a diffeomorphism, in particular injective. Let $v \in S_{p} M$ , $t > 0$ and $q : = γ_{v} (t) = \exp_{p} (t v)$ . By [F6] (Hopf–Rinow) there is $w \in T_{p} M$ with $\exp_{p} (w) = q$ and $∣ w ∣_{g_{p}} = d_{g} (p, q)$ ; injectivity forces $w = t v$ , so $d_{g} (p, q) = ∣ w ∣_{g_{p}} = t$ . As $t > 0$ was arbitrary, the minimizing-time set $A_{p} (v)$ of [F5] contains every positive time, so its supremum is $c_{p} (v) = + \infty$ .

2.1F2F4F5F7step 1.1step 1.2given

The ball volume is the model volume. [F2, F4, F5, F7, step 1.1, step 1.2, given] Fix $p \in M$ and $r > 0$ . The ball $B (p, r)$ is open, hence Borel, so by [F7] the polar formula [F4] applies to $f = 1_{B (p, r)}$ : ${vol}_{g} (B (p, r)) = \int_{S_{p} M} \int_{0}^{c_{p} (v)} 1_{B (p, r)} (γ_{v} (t)) \det a_{v} (t) d t d σ_{p} (v) .$ For $0 < t < c_{p} (v)$ the minimizing property in [F5] gives $d_{g} (p, γ_{v} (t)) = t$ , so $1_{B (p, r)} (γ_{v} (t)) = 1_{{t < r}}$ , while step 1.1 gives $\det a_{v} (t) = {sn}_{k} (t)^{n - 1}$ . By step 1.2 the cut time is infinite, so the inner integral equals $\int_{0}^{\min {c_{p} (v), r}} {sn}_{k} (t)^{n - 1} d t = \int_{0}^{r} {sn}_{k} (t)^{n - 1} d t,$ a finite number independent of $v$ . Therefore the outer integral is the constant inner value times $σ_{p} (S_{p} M) = ω_{n - 1}$ by [F4, F2], and ${vol}_{g} (B (p, r)) = ω_{n - 1} \int_{0}^{r} {sn}_{k} (t)^{n - 1} d t = \int_{0}^{r} A_{k} (t) d t = V_{k} (r) = V_{k}^{⋆} (r),$ using $A_{k} = ω_{n - 1} {sn}_{k}^{n - 1}$ and $V_{k}^{⋆} = V_{k}$ for $k \leq 0$ from [F2].

3.1F1F8F11step 2.1

The flat case. [F1, F8, F11, step 2.1] Let $k = 0$ . Then ${sn}_{0} (t) = t$ by [F1], so step 2.1 gives ${vol}_{g} (B (p, r)) = V_{0} (r) = ω_{n - 1} \int_{0}^{r} t^{n - 1} d t .$ The power integral [F8] with $m = n - 1 \geq 1$ and $s = 0$ evaluates this as $ω_{n - 1} r^{n} / n$ . This is the polynomial flat volume $V_{0}$ ; Euclidean $n$ -space, of identically zero curvature and metrically complete, realizes the case $k = 0$ [F11].

3.2F1step 2.1

The hyperbolic case, closed form. [F1, step 2.1] Let $k = - a^{2}$ . Then $\sqrt{- k} = a$ , so ${sn}_{- a^{2}} (t) = \sinh (a t) / a$ by [F1], and step 2.1 gives ${vol}_{g} (B (p, r)) = V_{- a^{2}} (r) = ω_{n - 1} \int_{0}^{r} {(\frac{\sinh (a t)}{a})}^{n - 1} d t .$

4.1F9F10step 3.2given

Exponential growth in negative curvature. [F9, F10, step 3.2, given] Write $f (t) : = (\sinh (a t) / a)^{n - 1} \geq 0$ for $t \geq 0$ . If $t \geq 1 / a$ then $a t \geq 1$ , and [F9] gives $\sinh (a t) \geq e^{a t} / 4$ , hence $f (t) \geq \frac{e^{a (n - 1) t}}{(4 a)^{n - 1}} .$ Now let $r \geq 2 / a$ and $t \in [r / 2, r]$ . Then $t \geq 1 / a$ and $t \geq r / 2$ , so $e^{a (n - 1) t} \geq e^{a (n - 1) r / 2}$ because $n - 1 \geq 1$ and the exponential is strictly increasing by [F9]; consequently $f \geq e^{a (n - 1) r / 2} / (4 a)^{n - 1}$ on the whole interval $[r / 2, r]$ . Since $f$ is nonnegative, the additivity and monotonicity of the integral [F10] give $\int_{0}^{r} f \geq \int_{r / 2}^{r} f \geq \frac{r}{2} \cdot \frac{e^{a (n - 1) r / 2}}{(4 a)^{n - 1}},$ and multiplying by $ω_{n - 1} > 0$ and combining with step 3.2 yields $V_{- a^{2}} (r) \geq \frac{ω_{n - 1} r}{2 (4 a)^{n - 1}} e^{a (n - 1) r / 2}, r \geq \frac{2}{a} .$ The exponent rate $a (n - 1) / 2$ is strictly positive because $n \geq 2$ , so the hyperbolic volume grows at least exponentially, in contrast with the polynomial flat volume of step 3.1; the constants $n$ , $a$ and $ω_{n - 1}$ are fixed by the model.

5.1F11F12step 2.1step 3.1step 4.1given∎

Realizations, Bishop–Gromov consistency and edge cases. [F11, F12, step 2.1, step 3.1, step 4.1, given] Euclidean $n$ -space has zero curvature and the upper-half-space metric of parameter $a$ has curvature $- a^{2}$ , by [F11], so the two computed cases carry the flat and hyperbolic curvature normalizations (the general statement is proved for every $(M, g)$ satisfying the hypotheses). By [F12] a space form of curvature $k$ has $Ric = (n - 1) k g$ , so both cases satisfy the hypotheses of the Bishop–Gromov comparison [F12], whose conclusion ${vol}_{g} (B (p, r)) \leq V_{k}^{⋆} (r)$ is attained with equality by step 2.1. Thus the comparison is sharp on the model, and step 4.1 shows that on the negative-curvature side the volume is allowed to grow exponentially, whereas the flat volume $V_{0} (r) = ω_{n - 1} r^{n} / n$ of step 3.1 is polynomial: a Ricci lower bound $Ric \geq - (n - 1) a^{2} g$ does not force polynomial volume growth. The degenerate cases are excluded or explained: $a > 0$ and $r > 0$ are fixed positive numbers; the threshold $r = 2 / a$ is finite; $n \geq 2$ makes $n - 1 \geq 1$ , which is exactly what the exponential rate needs (for $n = 1$ the integrand is constant and the growth is linear); and $k = 0$ , $k < 0$ exhaust the stated nonpositive curvature. The lower bound is stated only for $r \geq 2 / a$ , while the closed formula of step 3.2 holds for every $r > 0$ ; no upper bound and no exact asymptotic is claimed. No choice beyond the inherited [A1] is used: the radial direction, the minimizing geodesic and the model functions are fixed or explicit, and the cut-time, polar-integration, Cartan–Hadamard and Hopf–Rinow interfaces carry exactly ${A C}_{ω}$ .

Source locator

Datar §§27.2 and 28.1, pp.200–209, records the model radial density ${sn}_{k}^{n - 1}$ , the polar volume element and the model ball volume in the flat and hyperbolic cases; Eschenburg §§4–5, pp.15–20, introduces the same model radial density and its integral. The computation above evaluates the model volume functions $V_{0}$ and $V_{- a^{2}}$ from the comparison functions, identifies them with the ball volumes of the flat and hyperbolic space forms by the polar formula and Cartan–Hadamard, and derives the explicit exponential lower bound for the hyperbolic volume from $\sinh x \geq e^{x} / 4$ for $x \geq 1$ .

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-02Open item page →

Toponogov comparison on a round sphere

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Fix $k > 0$ , put $R : = 1 / \sqrt{k}$ , let $n \geq 2$ , and give the round sphere $S_{R}^{n} : = {x \in R^{n + 1} : ⟨ x, x ⟩ = R^{2}}$ the Riemannian metric $g$ induced by the Euclidean inner product (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). By Round sphere model geometry and The round sphere has positive constant sectional curvature, the manifold $(S_{R}^{n}, g)$ is complete, connected and boundaryless with constant sectional curvature $k = 1 / R^{2}$ ; realize the model surface $M_{k}^{2}$ of Comparison triangle in the two dimensional space form as a round two-sphere of radius $R$ . Then, for data satisfying the hypotheses of the comparison theorems of Toponogov hinge comparison and Toponogov triangle comparison:

Hinge equality. If $σ_{1}, σ_{2}$ are unit-speed minimizing geodesics of lengths $a, b > 0$ starting at a common point $p$ , with included angle $θ \in [0, π]$ , endpoints $x, y$ and $c : = d_{g} (x, y)$ , and if the hinge datum is admissible, then $c = c_{k} (a, b, θ)$ : the hinge comparison is an equality.
Self-comparison and angle equality. If $x, y, z \in S_{R}^{n}$ are joined by minimizing geodesic segments with admissible side lengths $(a, b, c)$ , then each actual vertex angle equals the corresponding comparison angle, and the triangle is its own comparison triangle: its vertices lie in a round two-sphere of radius $R$ inside $S_{R}^{n}$ , and an isometry of that two-sphere onto the model $M_{k}^{2}$ carries the triangle, together with its three minimizing sides, to a comparison triangle with side lengths $(a, b, c)$ .
Side-point equality (diagnostic). For admissible hinge data and the points $u = σ_{1} (s)$ , $v = σ_{2} (t)$ with $s \in [0, a]$ , $t \in [0, b]$ , one has $d_{g} (u, v) = d_{k} (\overset{ˉ}{u}, \overset{ˉ}{v})$ for the corresponding points $\overset{ˉ}{u}, \overset{ˉ}{v}$ of a model hinge with the same lengths and included angle: the model attains equality in the corresponding-side-point comparison. This is a direct computation for the model and is not used as a proof of the general comparison theorem.

Here admissible means that the hypotheses of the respective comparison theorem are satisfied: for a hinge, $σ_{1}, σ_{2}$ are unit-speed minimizing geodesics with positive lengths and, since $k > 0$ , $a, b, c < π R, a + b + c < 2 π R;$ for a triangle, the three positive side lengths satisfy the strict triangle inequalities and the same two displayed bounds, so that a comparison triangle in $M_{k}^{2}$ exists. In general the two theorems give only the inequality $c \leq c_{k} (a, b, θ)$ and the angle inequality; the example shows that on the round sphere both are equalities, and it identifies the actual triangle with a comparison triangle through the standard isometry between a great two-sphere and the model.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1]; the curvature $k > 0$ and the radius $R = 1 / \sqrt{k}$ ; the dimension $n \geq 2$ ; the round sphere $S_{R}^{n}$ with its induced metric $g$ ; the model $M_{k}^{2}$ realized as a round two-sphere of radius $R$ ; and, for the three claims, the data: hinge geodesics $σ_{1}, σ_{2}$ with common initial point $p$ , lengths $a, b > 0$ , unit initial directions $v_{1}, v_{2}$ , included angle $θ \in [0, π]$ and endpoints $x, y$ at distance $c$ ; triangle vertices $x, y, z$ joined by minimizing geodesic segments with side lengths $a, b, c > 0$ ; and the side points $u = σ_{1} (s)$ , $v = σ_{2} (t)$ , $s \in [0, a]$ , $t \in [0, b]$ , all data assumed admissible in the sense of the Example.

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried here by the exponential-map, cut-locus and comparison interfaces cited below. The example selects no family of objects: the only selections below are single selections from the nonempty finite-dimensional sets specified in step 3.1.

[F1]

Round-sphere geometry (Round sphere model geometry, The round sphere has positive constant sectional curvature, Constant sectional curvature and space form, Riemannian distance on a connected manifold): $S_{R}^{n}$ is a nonempty, connected, boundaryless embedded $n$ -manifold in $R^{n + 1}$ with $T_{x} S_{R}^{n} = x^{⊥}$ , the induced metric is the restriction of the Euclidean inner product, it makes $S_{R}^{n}$ geodesically complete and hence metrically complete, and the Riemannian distance is the metric distance on this connected manifold. Its sectional curvature is constantly $1 / R^{2} = k$ , so $(S_{R}^{n}, g)$ is a complete, connected, boundaryless manifold with $K \geq k$ , and the two-dimensional model $M_{k}^{2}$ is the round two-sphere of radius $R$ .

[F2]

Explicit geodesics and the distance formula (Round sphere model geometry, Existence uniqueness and smooth dependence of geodesics, Principal inverse sine and inverse cosine): for $p \in S_{R}^{n}$ and a unit vector $v \in T_{p} S_{R}^{n}$ , the maximal geodesic with $γ (0) = p$ , $γ^{'} (0) = v$ is $γ_{p, v} (t) = \cos (t / R) p + R \sin (t / R) v, t \in R,$ and it has unit speed. For all $p, q \in S_{R}^{n}$ , $d_{g} (p, q) = R \arccos ⁣ (\frac{⟨ p, q ⟩}{R^{2}}),$ the argument lying in $[- 1, 1]$ by Cauchy–Schwarz (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs); in particular $d_{g} (p, γ_{p, v} (t)) = t$ for $0 \leq t \leq π R$ , and for every real $δ$ with $0 < δ < 2 π$ one has $\arccos (\cos δ) = \min {δ, 2 π - δ}$ , because for $δ \in (0, π]$ this is the inverse property of the principal inverse cosine and for $δ \in [π, 2 π)$ one uses $\cos (2 π - δ) = \cos δ$ with $2 π - δ \in (0, π]$ .

[F3]

Unique maximal geodesics (Existence uniqueness and smooth dependence of geodesics): for every $(p, v) \in T S_{R}^{n}$ there is exactly one maximal geodesic with initial data $(p, v)$ , and every geodesic segment with those initial data is its restriction.

[F4]

Model comparison data (Comparison triangle in the two dimensional space form, Toponogov hinge comparison, Toponogov triangle comparison): $c_{k} (A, B, θ)$ denotes the distance in $M_{k}^{2}$ between the endpoints of unit-speed geodesics of lengths $A, B > 0$ issuing from a common point with included angle $θ$ ; the number is independent of the choices made. For fixed $A, B > 0$ with $A, B < π R$ the map $c_{k} (A, B, \cdot)$ is continuous and strictly increasing on $[0, π]$ , it is the inverse of the comparison-angle function, $c_{k} (A, B, Φ_{A, B} (C)) = C for ∣ A - B ∣ < C < m (A, B),$ where $m (A, B) : = \min {A + B, 2 π R - A - B}$ , and its endpoint values are $c_{k} (A, B, 0) = ∣ A - B ∣$ and $c_{k} (A, B, π) = m (A, B)$ . The comparison angle at the vertex opposite the side $C$ between the sides $A$ and $B$ is the unique $\overset{ˉ}{α} \in (0, π)$ with $\cos \overset{ˉ}{α} = F_{A, B} (C) : = \frac{\cos (C / R) - \cos (A / R) \cos (B / R)}{\sin (A / R) \sin (B / R)},$ and a comparison triangle with side lengths $(A, B, C)$ exists, uniquely up to the isometries of $M_{k}^{2}$ , exactly when the positive side lengths satisfy the strict triangle inequalities and $A, B, C < π R$ , $A + B + C < 2 π R$ . Under their stated hypotheses the hinge comparison gives $C \leq c_{k} (A, B, θ)$ and the triangle comparison gives that each actual vertex angle is at least the corresponding comparison angle.

[F5]

Angles (Pointwise norm and angle from a riemannian metric): for nonzero tangent vectors in one tangent space, the angle is the unique $θ \in [0, π]$ with $\cos θ = g (u, w) / (∣ u ∣_{g} ∣ w ∣_{g})$ ; in particular for unit vectors $\cos θ = g (u, w)$ .

[F6]

Euclidean bilinearity, Cauchy–Schwarz and addition formulas (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, The addition formulas for sine and cosine): the Euclidean inner product is bilinear and symmetric, $∣ ⟨ u, w ⟩ ∣ \leq ∣ u ∣ ∣ w ∣$ with equality only for linearly dependent $u, w$ , and $\cos (x \pm y) = \cos x \cos y \mp \sin x \sin y$ .

[F7]

Isometries and orthonormal bases (Every finite-dimensional real or complex inner product space has an orthonormal basis, Riemannian isometry and local isometry, Riemannian isometries preserve length and distance): every finite-dimensional real inner product space has an orthonormal basis. A Riemannian isometry $F$ is a diffeomorphism with $F^{*} h = g$ , hence for tangent vectors $ξ, η$ one has $g_{x} (ξ, η) = h_{F (x)} (d F_{x} ξ, d F_{x} η)$ , so isometries preserve angles as defined in [F5]; they also preserve the lengths of curves and the distances between points. A linear isometry $L : V \to R^{3}$ of a three-dimensional subspace $V \subseteq R^{n + 1}$ satisfies $⟨ L u, L v ⟩ = ⟨ u, v ⟩$ , so its restriction $L ∣_{V \cap S_{R}^{n}}$ is a Riemannian isometry onto the round sphere $S_{R}^{2}$ .

Verification

technique · direct: the round sphere has explicit geodesics and an explicit distance formula, so each comparison quantity is computed on both sides from the same two formulas and the values are identified; an explicit linear isometry between a great two-sphere containing the triangle and the model realizes the triangle as its own comparison triangle

1.1A1F1F2F3F4given

Setup and explicit sides. By [F1] the sphere $(S_{R}^{n}, g)$ is complete, connected and boundaryless with constant curvature $k$ , so it satisfies the curvature hypothesis $K \geq k$ of both comparison theorems of [F4]. The two hinge legs and the three triangle sides have positive lengths below $π R$ by admissibility; the hinge endpoint distance $c$ may be zero. Let $σ$ be such a segment, with starting point $p : = σ (0)$ and initial unit vector $v : = σ^{'} (0)$ ; by [F3] it is the restriction to its interval of the maximal geodesic $γ_{p, v}$ , so [F2] gives $σ (t) = \cos (t / R) p + R \sin (t / R) v, 0 \leq t \leq L,$ where $L$ is the length of $σ$ , and $⟨ p, p ⟩ = R^{2}$ , $⟨ p, v ⟩ = 0$ , $⟨ v, v ⟩ = 1$ . In particular every endpoint of the given hinge and triangle data is of this form, and, for two points $x, y$ obtained this way from a common base point, the distance $d_{g} (x, y)$ is computed from the Euclidean inner product $⟨ x, y ⟩$ by the distance formula of [F2].

2.1F2F3F4F5F6step 1.1given

Hinge equality. Write $v_{1} : = σ_{1}^{'} (0)$ and $v_{2} : = σ_{2}^{'} (0)$ . By [F5] the included angle satisfies $\cos θ = ⟨ v_{1}, v_{2} ⟩$ , since $g$ is the Euclidean inner product. Step 1.1 gives $x = \cos (a / R) p + R \sin (a / R) v_{1}, y = \cos (b / R) p + R \sin (b / R) v_{2},$ so bilinearity and $⟨ p, p ⟩ = R^{2}$ , $⟨ p, v_{i} ⟩ = 0$ , $⟨ v_{i}, v_{i} ⟩ = 1$ give $⟨ x, y ⟩ = R^{2} (\cos (a / R) \cos (b / R) + \sin (a / R) \sin (b / R) \cos θ) .$ The distance formula of [F2] therefore yields $c = d_{g} (x, y) = R \arccos (\cos (a / R) \cos (b / R) + \sin (a / R) \sin (b / R) \cos θ) .$ On the model side, [F4] says that $c_{k} (a, b, θ)$ is independent of the choices made; choose a model hinge in the realization $M_{k}^{2} = S_{R}^{2}$ , i.e. a point $\overset{ˉ}{p} \in S_{R}^{2}$ and unit-speed geodesics ${\overset{ˉ}{σ}}_{1}, {\overset{ˉ}{σ}}_{2}$ from $\overset{ˉ}{p}$ of lengths $a, b$ with included angle $θ$ , and write ${\overset{ˉ}{v}}_{i} : = {\overset{ˉ}{σ}}_{i}^{'} (0)$ , so that $⟨ {\overset{ˉ}{v}}_{1}, {\overset{ˉ}{v}}_{2} ⟩ = \cos θ$ by [F5]. The formulas of [F2] hold on $S_{R}^{2}$ as well (they are stated for every $n \geq 2$ ), so the same computation with $\overset{ˉ}{p}, {\overset{ˉ}{v}}_{1}, {\overset{ˉ}{v}}_{2}$ in place of $p, v_{1}, v_{2}$ gives $c_{k} (a, b, θ) = R \arccos (\cos (a / R) \cos (b / R) + \sin (a / R) \sin (b / R) \cos θ) .$ The two displayed values are equal, so $c = c_{k} (a, b, θ)$ : the hinge comparison of [F4], which asserts $c \leq c_{k} (a, b, θ)$ for these data, is an equality. The computation makes no use of the strictness of the inequalities $a, b, c < π R$ and $a + b + c < 2 π R$ beyond the well-definedness of the model hinge, which [F4] supplies.

2.2F2F3step 1.1given

Minimizing segments below the diameter are unique. Let $σ, τ$ be unit-speed minimizing geodesics in $S_{R}^{n}$ from a point $p$ to a point $q$ with $0 < L : = d_{g} (p, q) < π R$ . By [F3] and the explicit formula of [F2], $σ (t) = \cos (t / R) p + R \sin (t / R) σ^{'} (0)$ and $τ (t) = \cos (t / R) p + R \sin (t / R) τ^{'} (0)$ for $0 \leq t \leq L$ . Evaluating at $t = L$ , where both curves meet $q$ , gives $\cos (L / R) p + R \sin (L / R) σ^{'} (0) = \cos (L / R) p + R \sin (L / R) τ^{'} (0) .$ Since $0 < L / R < π$ , one has $\sin (L / R) > 0$ , so $σ^{'} (0) = τ^{'} (0)$ and hence $σ = τ$ . Consequently, in admissible triangle data the minimizing geodesic segment between two vertices at distance $< π R$ is unique, and the actual angles of the Example are independent of the choice of minimizing sides.

2.3F2F4F5step 1.1given

Angle equality by the spherical law of cosines. At the vertex $x$ of the triangle let the sides to $y$ and $z$ have lengths $c$ and $b$ , so that the opposite side is $a = d_{g} (y, z)$ , and let $α \in [0, π]$ be the angle at $x$ , between the unit directions $v : = σ_{x y}^{'} (0)$ and $w : = σ_{x z}^{'} (0)$ . By [F5], $\cos α = ⟨ v, w ⟩$ , and step 1.1 gives $y = \cos (c / R) x + R \sin (c / R) v, z = \cos (b / R) x + R \sin (b / R) w .$ Bilinearity and $⟨ x, x ⟩ = R^{2}$ , $⟨ x, v ⟩ = ⟨ x, w ⟩ = 0$ , $⟨ v, v ⟩ = ⟨ w, w ⟩ = 1$ yield $⟨ y, z ⟩ = R^{2} (\cos (b / R) \cos (c / R) + \sin (b / R) \sin (c / R) \cos α) .$ On the other hand $a = d_{g} (y, z) < π R$ , so the distance formula of [F2] gives $⟨ y, z ⟩ = R^{2} \cos (a / R)$ . Equating the two expressions and dividing by $\sin (b / R) \sin (c / R)$ , which is nonzero because $0 < b, c < π R$ , gives $\cos α = \frac{\cos (a / R) - \cos (b / R) \cos (c / R)}{\sin (b / R) \sin (c / R)} = F_{b, c} (a),$ the model expression of [F4] with $(A, B, C) = (b, c, a)$ . By [F4] the comparison angle $\overset{ˉ}{α} \in (0, π)$ opposite the side $a$ is the unique element of $(0, π)$ with $\cos \overset{ˉ}{α} = F_{b, c} (a)$ , while $α \in [0, π]$ by [F5]; cosine is injective on $[0, π]$ , so $α = \overset{ˉ}{α}$ . Cycling the roles of the vertices gives the equality at $y$ and at $z$ as well, with $β = \overset{ˉ}{β}$ and $γ = \overset{ˉ}{γ}$ .

3.1F1F2F4F7step 1.1step 2.3given

The triangle is its own comparison triangle. Let $W : = span {x, y, z} \subseteq R^{n + 1}$ . The distinct vertices $x, y$ are not antipodal because $0 < d_{g} (x, y) < π R$ , so they are linearly independent; hence $2 \leq \dim W \leq 3$ . There is a three-dimensional subspace $V$ with $W \subseteq V$ : if $\dim W = 3$ take $V = W$ ; otherwise $W^{⊥}$ is nonzero because $\dim W^{⊥} = n + 1 - \dim W \geq 3 - 2 > 0$ , and we take $V = W \oplus R u$ for one nonzero $u \in W^{⊥}$ . Then $x, y, z \in V$ , and each of the three minimizing sides lies in $V$ : by step 1.1 the side from a vertex $r$ to the other endpoint with unit direction $v$ is $t \mapsto \cos (t / R) r + R \sin (t / R) v$ , and this lies in $span {r, v} \subseteq V$ , since $v$ is determined by the two endpoints. By [F7] choose an orthonormal basis $(e_{1}, e_{2}, e_{3})$ of $V$ and define $L : V \to R^{3}$ by $L (α_{1} e_{1} + α_{2} e_{2} + α_{3} e_{3}) = (α_{1}, α_{2}, α_{3})$ ; then $L$ is a linear isometry, $⟨ L u, L w ⟩ = ⟨ u, w ⟩$ for $u, w \in V$ . Its restriction $Φ : = L ∣_{V \cap S_{R}^{n}}$ maps the round two-sphere $V \cap S_{R}^{n}$ of radius $R$ bijectively onto $S_{R}^{2}$ and is a Riemannian isometry by [F7], because on both spheres the metric is the restriction of the ambient Euclidean inner product. Hence $Φ$ preserves distances, lengths and angles. For a minimizing side $σ$ of the triangle, with unit direction $v$ at its startpoint $r$ and length $L_{0}$ , the curve $Φ \circ σ$ satisfies $Φ \circ σ (t) = \cos (t / R) Φ (r) + R \sin (t / R) L (v), 0 \leq t \leq L_{0},$ where $L (v)$ is a unit tangent vector of $S_{R}^{2}$ at $Φ (r)$ . By [F2] applied to $S_{R}^{2}$ , this is the unit-speed geodesic of the model with those initial data, so its length is $L_{0}$ ; and its endpoints have model distance $L_{0}$ because $Φ$ preserves distances. It is therefore a minimizing geodesic segment of the model. Applying this to the three sides, the triple $(Φ (x), Φ (y), Φ (z))$ , together with their image segments, has pairwise model distances $a, b, c$ and is thus a comparison triangle with side lengths $(a, b, c)$ in the model $M_{k}^{2}$ . Since $Φ$ preserves angles, its angles are the actual angles $α, β, γ$ , which by step 2.3 are the comparison angles; and comparison triangles with these side lengths are unique up to isometries of $M_{k}^{2}$ by [F4]. Hence, under the identification of the great two-sphere $V \cap S_{R}^{n}$ with the model by the isometry $Φ$ , the actual triangle is its own comparison triangle.

3.2F2F4step 1.1step 2.1given

Side-point equality. Write $u = σ_{1} (s) = \cos (s / R) p + R \sin (s / R) v_{1}$ and $v = σ_{2} (t) = \cos (t / R) p + R \sin (t / R) v_{2}$ by step 1.1, with $v_{1}, v_{2}$ unit and $⟨ v_{1}, v_{2} ⟩ = \cos θ$ . The bilinear computation of step 2.1 with $s, t$ in place of $a, b$ gives $⟨ u, v ⟩ = R^{2} (\cos (s / R) \cos (t / R) + \sin (s / R) \sin (t / R) \cos θ),$ so the distance formula of [F2] gives $d_{g} (u, v) = R \arccos (\cos (s / R) \cos (t / R) + \sin (s / R) \sin (t / R) \cos θ) .$ On the model side choose any point $\overset{ˉ}{p} \in S_{R}^{2} = M_{k}^{2}$ and unit vectors ${\overset{ˉ}{v}}_{1}, {\overset{ˉ}{v}}_{2} \in T_{\overset{ˉ}{p}} S_{R}^{2}$ with $⟨ {\overset{ˉ}{v}}_{1}, {\overset{ˉ}{v}}_{2} ⟩ = \cos θ$ ; such vectors exist because the tangent plane is two-dimensional. Put $\overset{ˉ}{u} : = \cos (s / R) \overset{ˉ}{p} + R \sin (s / R) {\overset{ˉ}{v}}_{1}$ and $\overset{ˉ}{v} : = \cos (t / R) \overset{ˉ}{p} + R \sin (t / R) {\overset{ˉ}{v}}_{2}$ , the model points at distances $s$ and $t$ along model geodesics of lengths $a$ and $b$ enclosing the angle $θ$ . The same computation in $S_{R}^{2}$ gives $d_{k} (\overset{ˉ}{u}, \overset{ˉ}{v}) = R \arccos (\cos (s / R) \cos (t / R) + \sin (s / R) \sin (t / R) \cos θ),$ an expression depending only on $s, t, θ$ and hence independent of the choices made. Therefore $d_{g} (u, v) = d_{k} (\overset{ˉ}{u}, \overset{ˉ}{v})$ for every $s \in [0, a]$ and $t \in [0, b]$ , including the endpoint choices $s = 0$ , $t = 0$ , $s = a$ and $t = b$ ; the case $s = a$ , $t = b$ reduces to step 2.1. This is the equality case of the corresponding-side-point comparison in the model, verified by direct computation; it is recorded as a diagnostic of the model geometry and is not used to prove the general corresponding-side-point comparison.

4.1A1F2F4F6step 1.1step 2.1step 2.2step 2.3step 3.1step 3.2given∎

Conclusion and boundary cases. [A1, F2, F4, F6, step 1.1, step 2.1, step 2.2, step 2.3, step 3.1, step 3.2, given] Steps 2.1, 2.3 and 3.2 show that on the round sphere of curvature $k$ the hinge, angle and side-point comparisons are equalities whenever their data are admissible, and step 3.1 exhibits the admissible triangle as its own comparison triangle under the isometry $Φ$ ; step 2.2 shows that the minimizing sides used are unique. The endpoint angles are covered by the same formula: if $θ = 0$ then $v_{2} = v_{1}$ by [F5] and [F6], and step 2.1 gives $c = R \arccos (\cos ((a - b) / R)) = ∣ a - b ∣ = c_{k} (a, b, 0)$ by [F2] and the endpoint value of [F4]; if $θ = π$ then $v_{2} = - v_{1}$ and step 2.1 gives $c = R \arccos (\cos ((a + b) / R)) = \min {a + b, 2 π R - a - b} = c_{k} (a, b, π)$ , because $0 < a + b < 2 π R$ and by the arccos identity of [F2] and the addition formulas of [F6]. The endpoint parameters $s, t \in {0, a} \times {0, b}$ were included in step 3.2. Triangle admissibility includes strict triangle inequalities and excludes collinear configurations. Hinge admissibility permits $θ = 0$ or $θ = π$ and hence collinear model hinges; these are covered by the endpoint formulas above, including $c = 0$ when $a = b$ and $θ = 0$ . Both kinds of data exclude sides equal to $π R$ and perimeter equal to $2 π R$ . No compactness of anything is assumed and no iff statement is made. On choice: the only selections are the single vector $u \in W^{⊥}$ and the single orthonormal basis of the three-dimensional space $V$ in step 3.1, each a selection from one nonempty set and not from a family; the inherited ${A C}_{ω}$ of [A1] suffices, and no full choice principle is used. Thus every admissible minimizing triangle of the round sphere is its own constant- $k$ comparison triangle, and all hinge, angle and side-point inequalities of the two comparison theorems become equalities.

Source locator

Lang, Riemannian and Metric Geometry, Chapter 5, Lemmas 5.1–5.2 and Theorem 5.15 (printed pp.64–70, PDF pp.67–73), gives the model cosine law, hinge monotonicity and Toponogov comparison whose equality case is exhibited here; Eschenburg, Comparison Theorems in Riemannian Geometry, §6, Theorem 6.1 and its Corollary 6.3, printed pp.21–25, proves the distance and angle comparisons for $K \geq k$ , the inequality directions used above. The computations are local: the geodesics and the distance formula are those of Round sphere model geometry, the model comparison data are those of Comparison triangle in the two dimensional space form, and the equality statements are verified on the explicit formulas rather than imported from the sources.

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Positive sectional curvature with no fixed lower bound on a noncompact manifold

Statement refuted

False claim: every complete, noncompact Riemannian manifold of everywhere positive sectional curvature has sectional curvature bounded below by a positive constant, $\inf K > 0$ .

The claim is false already in the simplest noncompact model surface: the paraboloid $P : = {(x, y, z) \in R^{3} : z = x^{2} + y^{2}}$ with the Riemannian metric induced from Euclidean $R^{3}$ is complete and noncompact, has positive sectional curvature at every point, and its sectional curvature takes the values $K = \frac{4}{(1 + 4 r^{2})^{2}} > 0, r^{2} = x^{2} + y^{2},$ at the point $(x, y, x^{2} + y^{2})$ , so $\inf_{P} K = 0$ : there is no positive lower bound.

Facts & Assumptions

Given: The paraboloid $P \subseteq R^{3}$ with the metric induced by the Euclidean metric, and the inherited ${A C}_{ω}$ of [A1].

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the hypersurface curvature and submanifold-completeness suppliers used below; the computation itself selects nothing.

[F1]

For an embedded Euclidean hypersurface of dimension $m \geq 2$ with a smooth unit normal and orthonormal principal directions $e_{i}, e_{j}$ of principal curvatures $κ_{i}, κ_{j}$ , the sectional curvature of $span (e_{i}, e_{j})$ is $κ_{i} κ_{j}$ (Euclidean hypersurface sectional curvature from principal curvatures). The principal curvatures are the eigenvalues of the shape operator $S$ , so in dimension $2$ the only sectional curvature is $K = \det S = κ_{1} κ_{2}$ (Shape operator, Principal curvatures, Gaussian curvature, and mean curvature of an oriented hypersurface).

[F2]

For a parametrized surface with first fundamental form $g = (g_{i j})$ and second fundamental form $h = (h_{i j})$ , the identity $h_{i j} = g (S X_{i}, X_{j})$ gives $S = g^{- 1} h$ in any coordinate basis, so $\det S = \det h / \det g$ (Induced connection and second fundamental form, Shape operator). For the graph of a smooth $f$ over the $(x, y)$ -plane with unit normal $N = (- f_{x}, - f_{y}, 1) / W$ , $W = \sqrt{1 + ∣ \nabla f ∣^{2}}$ , one has $g_{i j} = δ_{i j} + f_{i} f_{j}$ , $\det g = 1 + ∣ \nabla f ∣^{2}$ , and $h_{i j} = f_{i j} / W$ .

[F3]

The graph of a smooth map is an embedded submanifold (The graph of a smooth map is an embedded submanifold), and the graph of a continuous map into a Hausdorff space is closed in the product (The graph of a continuous map into a Hausdorff space is closed in the product).

[F4]

A closed embedded submanifold of a Riemannian manifold whose connected components are complete is complete in the induced Riemannian metric (Closed embedded submanifolds of complete Riemannian manifolds are complete); $R^{3}$ with the Euclidean metric is a complete metric space ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ , Complete metric space: every Cauchy sequence converges in the space), and its Riemannian distance is the Euclidean distance (Riemannian distance on a connected manifold).

[F5]

A metric space is compact exactly when every sequence in it has a convergent subsequence (Open cover, subcover, compact metric space, and compact subset of a metric space); a convergent sequence in $R^{3}$ is bounded. The infimum of a nonempty set of reals bounded below is its greatest lower bound (Greatest lower bound (infimum)).

Counterexample

1.1F3F4given

The paraboloid is a complete, noncompact surface. [F3, F4, given] The map $f : R^{2} \to R$ , $f (x, y) = x^{2} + y^{2}$ , is smooth and continuous, so its graph $P = {(x, y, f (x, y))}$ is an embedded submanifold of $R^{2} \times R ≅ R^{3}$ by [F3] and is closed in $R^{3}$ by [F3]. The Euclidean metric of $R^{3}$ is complete [F4], and its Riemannian distance is the Euclidean distance; hence [F4] makes $P$ complete in the induced Riemannian metric. For noncompactness consider $p_{t} : = (t, 0, t^{2}) \in P$ for $t = 1, 2, \dots$ : the Euclidean norms $∣ p_{t} ∣^{2} = t^{2} + t^{4}$ are unbounded, so $(p_{t})$ has no convergent subsequence and [F5] shows that $P$ is not compact.

2.1F1F2step 1.1

The curvature at each point. [F1, F2, step 1.1] Parametrize $P$ by $X (x, y) = (x, y, f (x, y))$ , so that $X_{x} = (1, 0, 2 x)$ , $X_{y} = (0, 1, 2 y)$ and $g = (\begin{array}{cc} 1 + 4 x^{2} & 4 x y \\ 4 x y & 1 + 4 y^{2} \end{array}), \det g = 1 + 4 (x^{2} + y^{2}) = 1 + 4 r^{2},$ where $r^{2} = x^{2} + y^{2}$ . The upward unit normal is $N = (1 + 4 r^{2})^{- 1 / 2} (- 2 x, - 2 y, 1)$ , and the second fundamental form has matrix $h = (1 + 4 r^{2})^{- 1 / 2} Hess f = (1 + 4 r^{2})^{- 1 / 2} (\begin{array}{cc} 2 & 0 \\ 0 & 2 \end{array}), \det h = \frac{4}{1 + 4 r^{2}} .$ By [F2] the shape operator satisfies $S = g^{- 1} h$ and $\det S = \frac{\det h}{\det g} = \frac{4 / (1 + 4 r^{2})}{1 + 4 r^{2}} = \frac{4}{(1 + 4 r^{2})^{2}} .$ By [F1] the sectional curvature of the tangent plane at $X (x, y)$ , the only tangent two-plane of the surface, is $K = \det S = 4 / (1 + 4 r^{2})^{2}$ .

3.1F5step 2.1∎

The curvature is positive but its infimum is zero. [F5, step 2.1] For every $(x, y)$ the numerator $4$ and the denominator $(1 + 4 r^{2})^{2}$ are positive, so $K = 4 / (1 + 4 r^{2})^{2} > 0$ : the paraboloid has positive sectional curvature everywhere. On the other hand, given any $δ > 0$ choose $r > 0$ with $(1 + 4 r^{2})^{2} > 4 / δ$ ; then the point $(r, 0, r^{2}) \in P$ has $0 < K < δ$ . Hence $0$ lies below every value of $K$ but no positive number is a lower bound, so the greatest lower bound of the set of values of $K$ is $\inf_{P} K = 0$ by [F5]: a uniform lower bound $K \geq c > 0$ fails on the complete noncompact manifold $P$ , refuting the displayed claim. The paraboloid and the chosen radii are explicit, so the inherited ${A C}_{ω}$ of [A1] is not drawn on beyond its declaration.

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Ricci lower bound does not control every sectional curvature in dimension at least three

Statement refuted

False claim: if a Riemannian manifold of dimension $n \geq 3$ satisfies a Ricci lower bound $Ric \geq (n - 1) k g$ , then every sectional curvature satisfies $K \geq k$ ; in particular the Ricci lower bound controls the individual sectional curvatures.

The product $M : = H^{2} (- 1) \times R^{n - 2}$ of the hyperbolic plane $H^{2} (- 1)$ of constant sectional curvature $- 1$ with the flat Euclidean factor, with the product metric, is a counterexample for every $n \geq 3$ :

$M$ is complete;
$Ric \geq - g = (n - 1) k g$ for $k : = - \frac{1}{n - 1}$ ;
a two-plane tangent to the $H^{2}$ factor at any point has $K = - 1 < k$ .

So a Ricci lower bound with $k < 0$ does not force the sectional curvature bound $K \geq k$ once $n \geq 3$ . Both factors are needed: on a surface Ricci is the sectional curvature, so no such failure occurs when $n = 2$ .

Facts & Assumptions

Given: The hyperbolic plane $H^{2} (- 1)$ realized as the upper half-plane $U^{2} = {y > 0}$ with $g_{1} = y^{- 2} (d x^{2} + d y^{2})$ , the Euclidean space $R^{n - 2}$ with the Euclidean metric and $n \geq 3$ , the product $M = H^{2} (- 1) \times R^{n - 2}$ with the product metric, and the inherited ${A C}_{ω}$ of [A1].

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the hyperbolic-completeness and product-completeness suppliers used below; the curvature and trace computations add no selection.

[F1]

Product manifolds and metrics (Products of smooth manifolds have a canonical product smooth structure, Canonical tangent and cotangent splittings for products, Coordinate criterion for a riemannian metric): the product of smooth manifolds carries its canonical product smooth structure, its tangent spaces split canonically as $T_{(p_{1}, p_{2})} (M_{1} \times M_{2}) ≅ T_{p_{1}} M_{1} \oplus T_{p_{2}} M_{2}$ , and a smooth symmetric positive-definite $(0, 2)$ -tensor field is a Riemannian metric; in a product chart $(x^{1}, \dots, x^{n_{1}}, y^{1}, \dots, y^{n_{2}})$ the product metric has the block-diagonal matrix $diag ((g_{1, i j} (x)), (g_{2, α β} (y)))$ , the first block depending only on $x$ and the second only on $y$ .

[F2]

Levi-Civita symbols and coordinate curvature (Fundamental theorem of riemannian geometry, Christoffel formula for the levi civita connection, Connection laws in directional form, Coordinate formula for the curvature tensor, Curvature is a type (1,3) tensor): the Levi-Civita connection is unique; in a chart its Christoffel symbols are $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{c d} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ , and the curvature components are $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b}$ ; the curvature is $C^{\infty}$ -linear in all three argument fields, so a formula verified on a coordinate frame holds for all tangent vectors.

[F3]

The two factors (Upper half-space model geometry, Euclidean space has zero curvature, $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ ): the upper half-plane $U^{2} = {y > 0}$ with $g_{1} = y^{- 2} (d x^{2} + d y^{2})$ is the case $a = 1$ , $n = 2$ of the half-space model, hence a complete Riemannian surface of constant sectional curvature $- 1$ ; Euclidean $R^{n - 2}$ has identically zero Riemann curvature and is metrically complete.

[F4]

Ricci is the trace $Ric (X, Y) = tr (Z \mapsto R (Z, X) Y)$ and, in an orthonormal basis $(e_{1}, \dots, e_{n})$ , $Ric (X, Y) = \sum_{i} Rm (e_{i}, X, Y, e_{i})$ , independently of the basis (Ricci curvature, Ricci curvature is symmetric and basis independent). For orthonormal $e_{i}, e_{j}$ , $Rm (e_{i}, e_{j}, e_{j}, e_{i}) = K (span (e_{i}, e_{j}))$ (Sectional curvature).

[F5]

Product completeness (A Riemannian product is complete iff each factor is complete): a finite Riemannian product is metrically complete exactly when each factor is metrically complete.

Counterexample

1.1F1F2given

The product connection splits. [F1, F2, given] In a product chart of [F1] the metric matrix is block diagonal, the first block a function of $x$ alone and the second a function of $y$ alone, and its inverse has the same block structure and dependence. In the Christoffel formula of [F2], if the upper index and the two lower indices are not all contained in the same block, then each of the three derivative terms either differentiates a mixed component, which vanishes identically, or differentiates a component of one block with respect to a coordinate of the other block, and each term $g^{a d}$ with $a$ in one block and $d$ in the other vanishes; hence $Γ^{a}_{b c} = 0$ whenever the three indices do not all lie in one block. When all three do lie in one block the formula is literally the Christoffel formula of that factor's metric.

2.1F1F2step 1.1

The curvature splits accordingly. [F1, F2, step 1.1] Insert the symbols of step 1.1 into the coordinate curvature formula of [F2]. If all four indices lie in the first block, then the symbols with all indices in that block are exactly those of $g_{1}$ and the first block of the inverse depends only on $x$ , so the formula reproduces the coordinate formula for the curvature of $g_{1}$ ; the same holds in the second block. If the indices meet both blocks, then every derivative term differentiates either an identically zero symbol or a block symbol with respect to a coordinate of the other block, and every quadratic term contains a symbol whose indices meet both blocks; so the component vanishes. By the tensoriality in [F2] and the canonical splitting of [F1], for tangent vectors decomposed accordingly, $R^{M_{1} \times M_{2}} ((u_{1}, u_{2}), (v_{1}, v_{2})) (w_{1}, w_{2}) = (R^{M_{1}} (u_{1}, v_{1}) w_{1}, R^{M_{2}} (u_{2}, v_{2}) w_{2}) .$ Consequently a two-plane inside the first factor has the sectional curvature computed from $g_{1}$ with the same Gram determinant, and a plane inside the second factor has the sectional curvature computed from $g_{2}$ ; in particular the planes inside the Euclidean factor are flat.

3.1F3step 2.1given

The curvature table of the product. [F3, step 2.1, given] Fix a point of $M$ and an orthonormal frame $e_{1}, \dots, e_{n}$ with $e_{1}, e_{2}$ tangent to the $H^{2}$ factor and $e_{3}, \dots, e_{n}$ tangent to the $R^{n - 2}$ factor. By [F3] the hyperbolic factor has constant sectional curvature $- 1$ and the Euclidean factor has identically zero curvature, so step 2.1 gives: $K (span (e_{1}, e_{2})) = - 1, K (span (e_{i}, e_{j})) = 0 otherwise,$ because every other pair of frame vectors either lies inside the flat factor, where [F3] gives vanishing curvature, or is mixed, in which case the curvature endomorphism of step 2.1 annihilates the pair.

4.1F4step 3.1

The Ricci tensor of the product. [F4, step 3.1] By the orthonormal-basis formula of [F4], the diagonal Ricci entries are $Ric (e_{j}, e_{j}) = \sum_{i \neq j} K (span (e_{i}, e_{j}))$ . Step 3.1 gives $Ric (e_{1}, e_{1}) = Ric (e_{2}, e_{2}) = - 1$ and $Ric (e_{j}, e_{j}) = 0$ for $j \geq 3$ . The curvature splitting of step 2.1 makes the mixed Ricci entries zero; on the two-dimensional hyperbolic factor the Ricci tensor is $- g_{1}$ , so its off-diagonal entry in the chosen orthonormal frame is also zero. Hence for $X = \sum_{j} x^{j} e_{j}$ , $Ric (X, X) = - (x^{1})^{2} - (x^{2})^{2} \geq - \sum_{j = 1}^{n} (x^{j})^{2} = - ∣ X ∣^{2},$ with equality exactly on $span (e_{1}, e_{2})$ . Thus $Ric \geq - g$ , and since $k = - 1 / (n - 1)$ gives $(n - 1) k = - 1$ , this is exactly $Ric \geq (n - 1) k g$ .

5.1F3F5step 4.1∎

The sectional bound fails and $M$ is complete. [F3, F5, step 4.1] The two-plane tangent to the hyperbolic factor has $K = - 1 < - \frac{1}{n - 1} = k$ , because $n \geq 3$ makes $\frac{1}{n - 1} < 1$ . So the Ricci lower bound does not imply $K \geq k$ . For completeness: by [F5] the product is metrically complete exactly when both factors are, and the hyperbolic plane and Euclidean space are metrically complete by [F3]; hence $M$ is metrically complete. In $n = 2$ the same construction degenerates to $H^{2} (- 1)$ alone, and there Ricci is the sectional curvature, so the failure genuinely needs dimension at least three. The product, its factors and the two-plane are explicit, so the inherited ${A C}_{ω}$ of [A1] is not drawn on beyond its declaration.

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Equality cases as diagnostics for all comparison signs

Example

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Fix $k \in R$ and $n \geq 2$ , and let $M_{k}^{n}$ be the complete, simply connected $n$ -dimensional space form of constant sectional curvature $k$ , with two-dimensional model $M_{k}^{2}$ and distance $d_{k}$ (Constant sectional curvature and space form). The five comparison theorems of this pair degenerate to equalities in these models, and the strict signs of their inequality directions are read off the explicit formulas for ${sn}_{k}$ , ${ct}_{k}$ and the model density:

Equality in the constant- $k$ model. In $M_{k}^{n}$ the normal Jacobi field with $J (0) = 0$ , $D_{t} J (0) = E$ is $J (t) = {sn}_{k} (t) P_{t} E$ , so the Rauch comparison is an equality; the Hessian and Laplace–Beltrami comparison bounds are attained with equality, $Hess r = {ct}_{k} (t_{0}) (g - d r \otimes d r)$ and $Δ_{g} r = (n - 1) {ct}_{k} (t_{0})$ ; the Bishop–Gromov ratio equals $1$ ; and in $M_{k}^{2}$ every admissible hinge has opposite side exactly $c_{k} (a, b, θ)$ while every admissible triangle is isometric to its own comparison triangle, so the hinge and triangle comparisons are equalities.
Strict signs of the flat-versus-curved models. For fixed admissible data and $k_{1} < k_{2}$ , the model quantities are strictly ordered at every time and radius in their common positive domains: ${sn}_{k_{2}} (t) < {sn}_{k_{1}} (t), {ct}_{k_{2}} (t) < {ct}_{k_{1}} (t), V_{k_{2}}^{⋆} (r) < V_{k_{1}}^{⋆} (r),$ and the model angle opposite a fixed side is strictly larger, $α_{k_{2}} (a, b, c) > α_{k_{1}} (a, b, c)$ , equivalently the model opposite side is strictly shorter, $c_{k_{2}} (a, b, θ) < c_{k_{1}} (a, b, θ)$ for $θ \in (0, π)$ . Thus against the flat model $k = 0$ the positive model lies strictly below it for the Jacobi, Hessian, Laplacian and volume quantities and strictly above it for the Toponogov angle, and the negative model reverses each strict sign: exactly the directions in which the comparison theorems of this pair read "more curvature means shorter Jacobi fields, smaller distance Hessian and Laplacian, smaller volume, fatter triangles".

The computation is a diagnostic for the direction conventions of Rauch comparison theorem first form, Hessian comparison for distance under sectional curvature bounds, Laplacian comparison for distance under a ricci lower bound, Bishop gromov volume comparison, Toponogov hinge comparison and Toponogov triangle comparison; it supplies no later proof and is never a dependency. For strict Jacobi-field inequalities the matched initial derivative $E$ is nonzero; if $E = 0$ , both fields vanish identically at every curvature. All approximations, limits and strict inequalities below are proved, not estimated numerically.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1]; the curvature $k \in R$ and dimension $n \geq 2$ ; the model functions ${sn}_{k}, {cs}_{k}, {ct}_{k}$ of [F1] with their ODE and Wronskian identities [F2]; for [F4] an admissible side datum $(a, b, c)$ in the sense of the comparison-triangle definition, its comparison angle $θ \in (0, π)$ opposite the side $c$ between the sides $a$ and $b$ , and the halves $s : = (a + b) / 2$ , $u : = (a - b) / 2$ , $v : = c / 2$ ; for the hinge form a unit-speed model hinge with sides $a, b > 0$ and included angle $θ$ ; a point $p$ in the space form, a point $q$ off $p$ and off its cut locus with $t_{0} : = d_{g} (p, q) > 0$ , and a radius $r > 0$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the comparison suppliers of this page. This example selects no family of objects: the model geodesics, Jacobi fields and comparison triangles it mentions are supplied by the cited definitions and theorems, and every computation below is performed on the explicit formulas.

[F1]

Model functions (Comparison sine, cosine and cotangent functions): ${sn}_{0} (t) = t$ , ${sn}_{k} (t) = \sin (\sqrt{k} t) / \sqrt{k}$ for $k > 0$ and ${sn}_{k} (t) = \sinh (\sqrt{- k} t) / \sqrt{- k}$ for $k < 0$ ; ${cs}_{k} = {sn}_{k}^{'}$ is $1$ , $\cos (\sqrt{k} t)$ , $\cosh (\sqrt{- k} t)$ in the same three cases; ${ct}_{k} = {cs}_{k} / {sn}_{k}$ ; ${sn}_{k}$ is odd and ${cs}_{k}$ is even. On the positive domain of ${sn}_{k}$ , ${sn}_{k} > 0$ ; ${ct}_{k} > 0$ for $k \leq 0$ , while for $k > 0$ it changes sign at $π / (2 \sqrt{k})$ within $(0, π / \sqrt{k})$ .

[F2]

Model ODE and Wronskian (Model functions solve the constant curvature jacobi equation): ${sn}_{k}^{''} + k {sn}_{k} = 0$ , ${cs}_{k}^{''} + k {cs}_{k} = 0$ , ${cs}_{k}^{2} + k {sn}_{k}^{2} = 1$ on $R$ , with ${sn}_{k} (0) = 0$ , ${cs}_{k} (0) = 1$ , ${sn}_{k}^{'} (0) = 1$ , ${cs}_{k}^{'} (0) = 0$ .

[F3]

Addition formulas (The addition formulas for complex trigonometric and hyperbolic functions, Addition formulas, identities, parity, and derivatives of the hyperbolic functions): the sine, cosine, hyperbolic sine and hyperbolic cosine addition formulas hold; consequently, for all real $x, y$ and all $k$ , ${sn}_{k} (x + y) = {sn}_{k} (x) {cs}_{k} (y) + {cs}_{k} (x) {sn}_{k} (y), {cs}_{k} (x \pm y) = {cs}_{k} (x) {cs}_{k} (y) \mp k {sn}_{k} (x) {sn}_{k} (y),$ and ${sn}_{k} (2 x) = 2 {sn}_{k} (x) {cs}_{k} (x)$ , ${cs}_{k} (2 x) = 1 - 2 k {sn}_{k} (x)^{2}$ . Moreover $(\cot x)^{'} = - \csc^{2} x$ and $(\coth y)^{'} = - {csch}^{2} y$ on their domains (Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant, the hyperbolic identities and derivatives theorem).

[F4]

Model cosine law and comparison triangle (Comparison triangle in the two dimensional space form): for an admissible side datum $(a, b, c)$ in $M_{k}^{2}$ there is a comparison triangle, unique up to isometry of $M_{k}^{2}$ , and its angle $θ$ opposite the side $c$ between the sides $a$ and $b$ is determined by the model cosine law, which in the unified notation is $\cos θ = \frac{{cs}_{k} (c) - {cs}_{k} (a) {cs}_{k} (b)}{k {sn}_{k} (a) {sn}_{k} (b)}$ for $k \neq 0$ and $\cos θ = (a^{2} + b^{2} - c^{2}) / (2 a b)$ for $k = 0$ . Admissibility means $a, b, c > 0$ , the strict triangle inequalities, and when $k > 0$ also $a, b, c < π / \sqrt{k}$ and $a + b + c < 2 π / \sqrt{k}$ ; the angle lies in $(0, π)$ .

[F5]

Model Jacobi fields and Rauch comparison (Model jacobi fields in positive zero and negative curvature, Rauch comparison theorem first form): in a space form of curvature $k$ the normal Jacobi field with $J (0) = 0$ , $D_{t} J (0) = E$ is $J (t) = {sn}_{k} (t) P_{t} E$ ; Rauch's theorem compares two manifolds with $\sec_{M_{1}} \geq \sec_{M_{2}}$ and gives $∣ J_{1} ∣ \leq ∣ J_{2} ∣$ for matched initial data up to the first conjugate instant of $γ_{1}$ .

[F6]

Hessian comparison (Hessian comparison for distance under sectional curvature bounds): at $q$ off $p$ and the cut locus, with $t_{0} < π / \sqrt{k}$ when $k > 0$ , the curvature bound $K \geq k$ along the minimizing geodesic gives $Hess r \leq {ct}_{k} (t_{0}) g$ on the normal hyperplane, and the reverse curvature bound gives the reverse inequality.

[F7]

Laplacian comparison (Laplacian comparison for distance under a ricci lower bound): $Ric \geq (n - 1) k g$ gives $Δ_{g} r (q) \leq (n - 1) {ct}_{k} (t_{0})$ .

[F8]

Space-form equalities (Distance hessian and laplacian in space forms): in the space form of curvature $k$ , $Hess r = {ct}_{k} (t_{0}) (g - d r \otimes d r)$ on all of $T_{q} M$ and $Δ_{g} r (q) = (n - 1) {ct}_{k} (t_{0})$ ; both comparison hypotheses of [F6] and [F7] hold with equality there.

[F9]

Bishop–Gromov comparison and model volumes (Bishop gromov volume comparison, Model space radial area and ball volume, Bishop gromov ratio is constant in the model space): for $Ric \geq (n - 1) k g$ the ratio $R_{p} (r) = {vol}_{g} (B (p, r)) / V_{k}^{⋆} (r)$ is nonincreasing with $\lim_{r ↓ 0} R_{p} (r) = 1$ ; the model radial area is $A_{k} (t) = ω_{n - 1} {sn}_{k} (t)^{n - 1}$ , the model ball volume is $V_{k} (r) = \int_{0}^{r} A_{k}$ , and in the space form of curvature $k$ , ${vol}_{g} (B (p, r)) = V_{k}^{⋆} (r)$ for every $r > 0$ .

[F10]

Toponogov hinge and triangle comparison (Toponogov hinge comparison, Toponogov triangle comparison): $K \geq k$ gives $c \leq c_{k} (a, b, θ)$ for every admissible minimizing hinge, where $c_{k} (a, b, θ)$ is the opposite side of the model hinge in $M_{k}^{2}$ determined by the cosine law of [F4]; and every actual angle of an admissible minimizing geodesic triangle is at least its comparison angle.

[F11]

Taylor–Peano (Peano's form: the normalized Taylor remainder tends to zero): a function that is $m$ times differentiable on an open neighborhood of $0$ has its order- $m$ Taylor expansion with remainder $o (x^{m})$ there. In particular the smooth sine and hyperbolic sine satisfy $\sin x = x - x^{3} / 6 + o (x^{3})$ and $\sinh x = x + x^{3} / 6 + o (x^{3})$ .

Verification

technique · direct. The comparison coefficients of the five theorems are explicit functions of $k$, so equality in the model and the strict signs are computed from the displayed formulas. The only non-elementary point is the strict monotonicity of the model angle in $k$: differentiating the half-angle form of the model cosine law writes the derivative of the angle as a positive multiple of $\cos^2(\theta/2)G(u)+\sin^2(\theta/2)G(s)-G(v)$, where $G(t)=\operatorname{sn}_k(t)m_k(t)$ with $m_k(t)=\int_0^t x\operatorname{sn}_k(x)\,dx$, and this expression is positive because $t\mapsto G(t)$ is, as a function of $h=\operatorname{sn}_k^{\,2}$, a strictly convex function

1.1F3

Elementary scalar estimates. [F3] For $x > 0$ one has $\sin x < x$ : the function $x - \sin x$ vanishes at $0$ and has derivative $1 - \cos x \geq 0$ , which is positive except at multiples of $2 π$ , so it is strictly increasing on $[0, 2 π]$ and then visibly positive; for $x > 0$ one has $\sinh x > x$ , since $\sinh x - x$ vanishes at $0$ and has derivative $\cosh x - 1 \geq 0$ , positive for $x > 0$ . For $0 < x < π$ one has $\cot x < 1 / x$ : for $0 < x \leq π / 2$ the function $\sin x - x \cos x$ vanishes at $0$ and has derivative $x \sin x > 0$ , so $\sin x > x \cos x$ and $\cot x = \cos x / \sin x < 1 / x$ ; for $π / 2 < x < π$ one has $\cot x < 0 < 1 / x$ . For $y > 0$ one has $\coth y > 1 / y$ : $y \cosh y - \sinh y$ vanishes at $0$ and has derivative $y \sinh y > 0$ , so $y \cosh y > \sinh y$ and $\coth y = \cosh y / \sinh y > 1 / y$ . Also $\cos x \leq 1$ and $\cosh y \geq 1$ , with the latter inequality strict for $y > 0$ .

1.2F1F2F5F11step 1.1

The model Jacobi field is strictly shorter at larger curvature. [F1, F2] For fixed $t > 0$ put $m_{k} (t) : = \int_{0}^{t} x {sn}_{k} (x) d x .$ Since ${sn}_{k} > 0$ on $(0, D_{k})$ , where $D_{k} : = π / \sqrt{k}$ for $k > 0$ and $D_{k} : = + \infty$ for $k \leq 0$ , one has $m_{k} (t) > 0$ for $0 < t < D_{k}$ . From the explicit formulas, for $k \neq 0$ , $\partial_{k} {sn}_{k} (t) = \frac{t {cs}_{k} (t) - {sn}_{k} (t)}{2 k} = - \frac{m_{k} (t)}{2},$ the first equality following by differentiating the displayed quotient $\sin (\sqrt{k} t) / \sqrt{k}$ (respectively $\sinh (\sqrt{- k} t) / \sqrt{- k}$ ) in $k$ , and the second from $t {cs}_{k} - {sn}_{k} = (t {cs}_{k} - {sn}_{k}) (0) + \int_{0}^{t} (- k x {sn}_{k} (x)) d x = - k m_{k} (t)$ , which uses $({cs}_{k})^{'} = - k {sn}_{k}$ [F2]. At $k = 0$ , [F11] gives ${sn}_{k} (t) = t - k t^{3} / 6 + o (k)$ from either side, so $\partial_{k} {sn}_{k} (t) ∣_{k = 0} = - t^{3} / 6 = - m_{0} (t) / 2$ . The derivative formula extends continuously across $k = 0$ since the explicit model functions converge uniformly on the finite interval $[0, t]$ . Hence $\partial_{k} {sn}_{k} (t) < 0$ : for fixed $t > 0$ the function $k \mapsto {sn}_{k} (t)$ is continuously differentiable and strictly decreasing on $(- \infty, π^{2} / t^{2})$ . In particular ${sn}_{k} (t) < t$ for $0 < k < π^{2} / t^{2}$ , ${sn}_{0} (t) = t$ and ${sn}_{k} (t) > t$ for $k < 0$ ; by [F5] the same comparison is the strict sign of the Rauch comparison between the space forms of curvatures $k_{1} < k_{2}$ : for matched initial derivatives of the same norm $∣ E ∣ > 0$ , one has $∣ J_{k_{2}} (t) ∣ = {sn}_{k_{2}} (t) ∣ E ∣ < {sn}_{k_{1}} (t) ∣ E ∣ = ∣ J_{k_{1}} (t) ∣$ . If $E = 0$ , both fields are identically zero.

1.3F1F3F6F7F8step 1.1

The model cotangent is strictly smaller at larger curvature. [F1, F3] From the explicit formulas, ${ct}_{0} (t) = 1 / t$ , ${ct}_{k} (t) = \sqrt{k} \cot (\sqrt{k} t)$ for $k > 0$ and ${ct}_{k} (t) = \sqrt{- k} \coth (\sqrt{- k} t)$ for $k < 0$ , and ${ct}_{k} (t) \to 1 / t$ as $k \to 0$ in either sign. For $k > 0$ , with $x = \sqrt{k} t \in (0, π)$ , $\frac{d}{d k} {ct}_{k} (t) = \frac{\sin x \cos x - x}{2 \sqrt{k} \sin^{2} x} < 0,$ because $\sin x \cos x \leq \sin x < x$ for $x > 0$ by step 1.1 (for $x \leq π / 2$ use $\sin x < x$ and $\cos x \leq 1$ ; for $x > π / 2$ , $\sin x \cos x < 0 < x$ ). For $k < 0$ , with $y = \sqrt{- k} t > 0$ , $\frac{d}{d k} {ct}_{k} (t) = - \frac{\sinh y \cosh y - y}{2 \sqrt{- k} \sinh^{2} y} < 0,$ because $y < \sinh y \cosh y$ by step 1.1 applied to $\sinh (2 y) > 2 y$ . Hence for fixed $t > 0$ the function $k \mapsto {ct}_{k} (t)$ is strictly decreasing on $(- \infty, π^{2} / t^{2})$ , and ${ct}_{k} (t) < 1 / t$ for $k > 0$ , $= 1 / t$ for $k = 0$ , $> 1 / t$ for $k < 0$ . By [F8] this is exactly the strict sign of the Hessian and Laplacian comparisons between the model of curvature $k$ and the flat model: the curvature bound $K \geq k^{'}$ holds in $M_{k}^{n}$ whenever $k \geq k^{'}$ [F6, F7], and the attained model value ${ct}_{k}$ is strictly below ${ct}_{k^{'}}$ when $k > k^{'}$ .

1.4F4F5F8F9F10

Equality in the constant- $k$ model. The five comparisons are equalities in $M_{k}^{n}$ and $M_{k}^{2}$ :

Jacobi: by [F5] the matched normal Jacobi field of the model is $J (t) = {sn}_{k} (t) P_{t} E$ , so the Rauch bound is attained at every $t$ of the domain.
Hessian and Laplacian: by [F8], $Hess r$ and $Δ_{g} r$ equal the comparison expressions of [F6] and [F7] exactly, and the two curvature hypotheses hold with equality.
Volume: by [F9], ${vol}_{g} (B (p, r)) = V_{k}^{⋆} (r)$ , so the Bishop–Gromov ratio is identically $1$ .
Toponogov: by [F4] a comparison triangle in $M_{k}^{2}$ is unique up to isometry, so the comparison triangle of an admissible model triangle is congruent to the triangle itself and every model angle equals its comparison angle; in the hinge form the model opposite side is by definition $c_{k} (a, b, θ)$ [F10], so both comparisons hold with equality. In each case the bounding quantity is exactly the model quantity, so no strictness remains inside a fixed model; strictness can only appear between models of different curvature, which is what steps 1.2, 1.3, 2.1 and 5.1 compute.

1.5F1

The primitives $m$ , $G$ and $ρ$ . [F1] For the rest of the computation fix $k$ and write $m (t) : = m_{k} (t) = \int_{0}^{t} x {sn}_{k} (x) d x$ on the positive domain, $h (t) : = {sn}_{k} (t)^{2}$ , $G (t) : = {sn}_{k} (t) m (t) = h (t) ρ (t)$ with $ρ (t) : = m (t) / {sn}_{k} (t)$ , and $\tan_{k} (t) : = {sn}_{k} (t) / {cs}_{k} (t)$ . Then $m (0) = 0$ , $m^{'} = t {sn}_{k} (t)$ , $m$ is odd and $m > 0$ on $(0, D_{k})$ ; $h$ is even and strictly increasing on $(0, D_{k} / 2)$ (for $k > 0$ its restriction is $\sin^{2} (\sqrt{k} t) / k$ , for $k \leq 0$ it is $t^{2}$ or $\sinh^{2} (\sqrt{- k} t) / (- k)$ ), with $h (t) > 0$ for $0 < t < D_{k}$ ; $G$ is even and $G > 0$ on $(0, D_{k})$ ; and on $(0, D_{k} / 2)$ one has ${cs}_{k} > 0$ , hence $\tan_{k} > 0$ .

1.6F3F4step 1.5

Half-angle form of the model cosine law. [F3, F4] Let $(a, b, c)$ be admissible with angle $θ \in (0, π)$ opposite $c$ as in [F4], and put $s = (a + b) / 2$ , $u = (a - b) / 2$ , $v = c / 2$ . Then $\sin^{2} \frac{θ}{2} = \frac{h (v) - h (u)}{{sn}_{k} (a) {sn}_{k} (b)}, \cos^{2} \frac{θ}{2} = \frac{h (s) - h (v)}{{sn}_{k} (a) {sn}_{k} (b)},$ and consequently the two identities $h (s) - h (u) = {sn}_{k} (a) {sn}_{k} (b), h (v) = \cos^{2} \frac{θ}{2} h (u) + \sin^{2} \frac{θ}{2} h (s) .$ For $k \neq 0$ this is a computation from the cosine law of [F4]: the addition formulas of [F3] give ${cs}_{k} (a \pm b) = {cs}_{k} (a) {cs}_{k} (b) \mp k {sn}_{k} (a) {sn}_{k} (b)$ , so $1 - \cos θ = \frac{{cs}_{k} (a - b) - {cs}_{k} (c)}{k {sn}_{k} (a) {sn}_{k} (b)}, 1 + \cos θ = \frac{{cs}_{k} (c) - {cs}_{k} (a + b)}{k {sn}_{k} (a) {sn}_{k} (b)},$ and ${cs}_{k} (2 x) = 1 - 2 k {sn}_{k} (x)^{2}$ converts the right hand sides into the two displayed expressions, whose sum is $1$ and gives the third identity; the fourth follows by substituting the third into the first. For $k = 0$ the same identities are the elementary Euclidean computation: $\sin^{2} (θ / 2) = (v^{2} - u^{2}) / (a b)$ , $\cos^{2} (θ / 2) = (s^{2} - v^{2}) / (a b)$ , $s^{2} - u^{2} = a b$ and $v^{2} = \cos^{2} (θ / 2) u^{2} + \sin^{2} (θ / 2) s^{2}$ follow from $c^{2} = a^{2} + b^{2} - 2 a b \cos θ$ . Also, the map $θ \mapsto c_{k} (a, b, θ)$ is strictly increasing on $(0, π)$ : the first identity writes $h (c_{k} (a, b, θ) / 2) = h (u) + {sn}_{k} (a) {sn}_{k} (b) \sin^{2} (θ / 2)$ whose right-hand side varies strictly increasingly in $θ$ . For $k > 0$ the model opposite side is less than $D_{k}$ , so its half lies in $(0, D_{k} / 2)$ , where $h$ is strictly increasing; for $k \leq 0$ the same holds for every positive half-side. Hence $c_{k} (a, b, θ)$ and $θ$ determine each other strictly increasingly.

2.1F1F9step 1.2

The saturated model volume is strictly smaller at larger curvature. [F1, F9, step 1.2] For $k_{1} < k_{2}$ , put $D_{i} : = D_{k_{i}}$ , with $D_{i} = + \infty$ when $k_{i} \leq 0$ . Then $D_{2} \leq D_{1}$ , and step 1.2 gives $A_{k_{2}} (t) < A_{k_{1}} (t)$ for $0 < t < D_{2}$ . For every $r > 0$ , $V_{k_{i}}^{⋆} (r) = \int_{0}^{\min {r, D_{i}}} A_{k_{i}} (t) d t .$ The $k_{1}$ integral covers at least $(0, \min {r, D_{2}})$ , on which its density is strictly larger. Hence $V_{k_{2}}^{⋆} (r) < V_{k_{1}}^{⋆} (r) (r > 0) .$ Thus the larger-curvature model has strictly smaller saturated balls, the strict form of the flat-versus-curved Bishop–Gromov comparison.

2.2F1F3step 1.1step 1.5

The ratio $ρ$ is strictly increasing. In the three cases, using the explicit forms of ${sn}_{k}$ and the identity $m (t) = ({sn}_{k} (t) - t {cs}_{k} (t)) / k$ for $k \neq 0$ (both sides vanish at $0$ and have derivative $t {sn}_{k} (t)$ ):

$k > 0$ : with $x = \sqrt{k} t$ , $ρ (t) = (1 - x \cot x) / k$ and $ρ^{'} (t) = (x - \sin x \cos x) / (\sqrt{k} \sin^{2} x) > 0$ by step 1.1;
$k = 0$ : $ρ (t) = t^{2} / 3$ , so $ρ^{'} (t) = 2 t / 3 > 0$ ;
$k < 0$ : with $y = \sqrt{- k} t$ , $ρ (t) = (y \coth y - 1) / (- k)$ and $ρ^{'} (t) = (\sinh y \cosh y - y) / (\sqrt{- k} \sinh^{2} y) > 0$ by step 1.1. Thus $ρ$ is strictly increasing on $(0, D_{k})$ . No monotonicity of $G = h ρ$ on the whole positive domain is asserted.

2.3F2F3step 1.5

The addition identity for $G$ . [F2, F3, step 1.5] For all $a, b > 0$ , with $s = (a + b) / 2$ and $u = (a - b) / 2$ , ${sn}_{k} (u) m (u) + \frac{1}{2} [m (a) {sn}_{k} (b) + {sn}_{k} (a) m (b)] = {sn}_{k} (s) m (s) .$ For $k \neq 0$ , multiply the difference of the two sides by $k$ and use $m (t) = ({sn}_{k} (t) - t {cs}_{k} (t)) / k$ together with the product-to-sum consequences of [F3]: ${sn}_{k} (x) {cs}_{k} (x) = {sn}_{k} (2 x) / 2$ , ${sn}_{k} (x) {sn}_{k} (y) = ({cs}_{k} (x - y) - {cs}_{k} (x + y)) / (2 k)$ , ${cs}_{k} (x) {cs}_{k} (y) = ({cs}_{k} (x - y) + {cs}_{k} (x + y)) / 2$ , ${sn}_{k} (x) {cs}_{k} (y) = ({sn}_{k} (x + y) + {sn}_{k} (x - y)) / 2$ ; then the difference equals $[h (u) - h (s) + {sn}_{k} (a) {sn}_{k} (b)] + [- \frac{u}{2} {sn}_{k} (2 u) - \frac{1}{2} (a {cs}_{k} (a) {sn}_{k} (b) + b {sn}_{k} (a) {cs}_{k} (b)) + \frac{s}{2} {sn}_{k} (2 s)],$ whose first bracket is zero by the addition formula for ${cs}_{k} (2 \cdot)$ and whose second bracket is zero because $a {cs}_{k} (a) {sn}_{k} (b) + b {sn}_{k} (a) {cs}_{k} (b) = s {sn}_{k} (2 s) - u {sn}_{k} (2 u)$ ; hence the difference is zero. For $k = 0$ the identity reads $u^{4} / 3 + (a^{3} b + a b^{3}) / 6 = s^{4} / 3$ and follows from $s^{4} - u^{4} = (s - u) (s + u) (s^{2} + u^{2}) = a b (a^{2} + b^{2}) / 2$ .

3.1F2step 1.5step 2.2

The conjugate function $Γ$ is strictly convex. [F2, step 2.2, step 1.5] Let $Γ$ be defined on the set of values of $h$ by $Γ (h (t)) : = G (t)$ for $t \in (0, D_{k} / 2)$ ; this is well defined because $h$ is a strictly increasing bijection from $(0, D_{k} / 2)$ onto its image. The formulas $G (t) \to 0$ and $h (t) \to 0$ as $t ↓ 0$ extend $Γ$ continuously by $Γ (0) = 0$ ; when $k > 0$ , the explicit model formulas likewise extend it continuously to $h (D_{k} / 2)$ . Then $\frac{d Γ}{d h} = \frac{ρ (t)}{2} + \frac{t}{2} \tan_{k} (t), \frac{d^{2} Γ}{d h^{2}} = \frac{ρ^{'} (t) + \tan_{k} (t) + t / {cs}_{k} (t)^{2}}{4 {sn}_{k} (t) {cs}_{k} (t)} > 0$ at $h = h (t)$ , $t \in (0, D_{k} / 2)$ . Indeed, $d h / d t = 2 {sn}_{k} {cs}_{k}$ and $d G / d t = {cs}_{k} m + {sn}_{k} m^{'} = {sn}_{k} ({cs}_{k} ρ + t {sn}_{k})$ , so $\frac{d Γ}{d h} = (d G / d t) / (d h / d t) = ρ / 2 + (t / 2) \tan_{k}$ ; differentiating the last expression with respect to $t$ and dividing by $2 {sn}_{k} {cs}_{k}$ gives the displayed second derivative, whose numerator is positive because $ρ^{'} > 0$ (step 2.2), $\tan_{k} > 0$ , $t > 0$ and ${sn}_{k}, {cs}_{k} > 0$ on $(0, D_{k} / 2)$ , while $(\tan_{k})^{'} = (1 + k \tan_{k}^{2}) = {cs}_{k}^{- 2}$ by the Wronskian identity [F2]. Hence $Γ$ is strictly convex on the interior and, by its continuous endpoint extension, on every closed subinterval of the extended domain.

4.1step 1.5step 1.6step 2.2step 2.3step 3.1

The half-angle numerator is positive. [step 1.5, step 1.6, step 2.2, step 2.3, step 3.1] Let $(a, b, c)$ be admissible, $θ \in (0, π)$ its angle at the vertex between the sides $a$ and $b$ , and set $M_{1} : = \cos^{2} \frac{θ}{2} G (u) + \sin^{2} \frac{θ}{2} G (s) - G (v), s = \frac{a + b}{2}, u = \frac{a - b}{2}, v = \frac{c}{2} .$ Then $M_{1} > 0$ . To see this, put $t_{s} : = s$ if $k \leq 0$ or $s \leq D_{k} / 2$ , and $t_{s} : = D_{k} - s$ if $k > 0$ and $s > D_{k} / 2$ . Then $t_{s} \in (0, D_{k} / 2]$ and $h (t_{s}) = h (s)$ . Also $∣ u ∣ < v < t_{s}$ : when $t_{s} = s$ this follows from the strict triangle inequality, and when $s > D_{k} / 2$ it follows from $a + b + c < 2 D_{k}$ . Since $ρ$ is strictly increasing and $h (s) = h (t_{s}) > 0$ , $G (t_{s}) \leq G (s)$ , strictly if $t_{s} < s$ . Thus $∣ u ∣ < v < t_{s} \leq D_{k} / 2$ , and $h (u), h (v), h (t_{s})$ lie in the increasing range of $h$ ; the fourth identity of step 1.6 makes $h (v)$ a strict convex combination of $h (u)$ and $h (t_{s})$ with weights $\cos^{2} (θ / 2)$ and $\sin^{2} (θ / 2)$ , both in $(0, 1)$ . Strict convexity of $Γ$ (step 3.1) on the closed interval between $h (u)$ and $h (t_{s})$ gives $G (v) = Γ (h (v)) < \cos^{2} \frac{θ}{2} Γ (h (u)) + \sin^{2} \frac{θ}{2} Γ (h (t_{s})) = \cos^{2} \frac{θ}{2} G (u) + \sin^{2} \frac{θ}{2} G (t_{s}) \leq \cos^{2} \frac{θ}{2} G (u) + \sin^{2} \frac{θ}{2} G (s),$ so $M_{1} > 0$ ; here $G (u) = G (∣ u ∣)$ because $G$ is even. This is the only place where the saturation $s > D_{k} / 2$ of the positive-curvature model is handled.

5.1F1F4step 1.2step 1.6step 4.1

The model angle is strictly increasing in the curvature. [F1, F4, step 1.2, step 1.6, step 4.1] Fix an admissible side datum $(a, b, c)$ and let $θ (k)$ be the model angle opposite $c$ between the sides $a$ and $b$ in $M_{k}^{2}$ , as $k$ ranges over the open admissible interval $(- \infty, k_{\max})$ (where $k_{\max} : = \min {(π / \max {a, b, c})^{2}, (2 π / (a + b + c))^{2}}$ ); on this interval the datum stays nondegenerate and $θ (k) \in (0, π)$ . Then $θ^{'} (k) = \frac{2 M_{1} (k)}{{sn}_{k} (a) {sn}_{k} (b) \sin θ (k)} > 0,$ where $M_{1} (k)$ is the positive quantity of step 4.1 computed with the model functions at $k$ . Indeed, differentiating the fourth identity of step 1.6 in $k$ gives $- G (v) = \frac{d}{d k} [\cos^{2} \frac{θ}{2}] (h (u) - h (s)) - \cos^{2} \frac{θ}{2} G (u) - \sin^{2} \frac{θ}{2} G (s),$ because at each fixed argument $t$ , $\partial_{k} h (t) = 2 {sn}_{k} (t) \partial_{k} {sn}_{k} (t) = - G (t)$ by step 1.2; since $\frac{d}{d k} \cos^{2} (θ / 2) = - \frac{1}{2} \sin θ θ^{'}$ and $h (u) - h (s) = - {sn}_{k} (a) {sn}_{k} (b)$ by step 1.6, this rearranges to the displayed formula. Its right-hand side is positive because $M_{1} (k) > 0$ , ${sn}_{k} (a), {sn}_{k} (b) > 0$ (the datum is admissible at $k$ ) and $\sin θ (k) > 0$ . Therefore $θ$ is strictly increasing on the admissible interval: for $k_{1} < k_{2}$ with the datum admissible at both, $α_{k_{2}} (a, b, c) = θ (k_{2}) > θ (k_{1}) = α_{k_{1}} (a, b, c)$ . For fixed $θ \in (0, π)$ , if $c_{k_{2}} (a, b, θ) \geq c_{k_{1}} (a, b, θ)$ , strict increase of the angle at fixed side lengths in $k$ and of the model side at fixed $k$ in $θ$ would give an angle greater than $θ$ at $k_{2}$ for its own side length, a contradiction. Thus $c_{k_{2}} (a, b, θ) < c_{k_{1}} (a, b, θ)$ .

6.1F5F6F7F8F9F10step 1.2step 1.3step 2.1step 1.4step 5.1

The five strict signs and their directions. Assemble the computations. For $k_{1} < k_{2}$ with the relevant data admissible at both curvatures:

Rauch: for matched initial derivatives with common norm $∣ E ∣ > 0$ , the model Jacobi field at $k_{2}$ is strictly shorter than at $k_{1}$ at every positive time in their common domain (step 1.2); in particular, against the flat model, positive curvature gives strictly shorter fields and negative curvature strictly longer fields, while inside a fixed model the bound is an equality (step 1.4). This is consistent with [F5]: in $M_{k_{2}}^{n}$ one has $\sec = k_{2} \geq k_{1}$ , so Rauch's inequality $∣ J_{k_{2}} ∣ \leq ∣ J_{k_{1}} ∣$ is strict for $E \neq 0$ ; for $E = 0$ both sides are zero.
Hessian and Laplacian: ${ct}_{k_{2}} (t) < {ct}_{k_{1}} (t)$ for $t > 0$ in the common domain (step 1.3), so the space-form values ${ct}_{k}$ of [F8] give strict inequalities relative to the comparison values of [F6], [F7]: the flat-versus-positive sign is ${ct}_{k} (t) < 1 / t$ and the flat-versus-negative sign is ${ct}_{k} (t) > 1 / t$ . The directions match $Hess r \leq {ct}_{k} g$ and $Δ_{g} r \leq (n - 1) {ct}_{k}$ under $K \geq k$ , $Ric \geq (n - 1) k g$ .
Volume: $V_{k_{2}}^{⋆} (r) < V_{k_{1}}^{⋆} (r)$ for every $r > 0$ (step 2.1): the model of larger curvature has strictly smaller balls, matching the Bishop–Gromov upper bound, with equality inside a fixed model (step 1.4).
Toponogov: $α_{k_{2}} > α_{k_{1}}$ and $c_{k_{2}} (a, b, θ) < c_{k_{1}} (a, b, θ)$ for nondegenerate data (step 5.1): the model of larger curvature has strictly fatter triangles and strictly shorter opposite sides, matching the hinge inequality $c \leq c_{k} (a, b, θ)$ and the angle inequality $α \geq \overset{ˉ}{α}$ under $K \geq k$ [F10], with equality inside a fixed model (step 1.4). All equalities inside a fixed model are those of step 1.4. The example is a direction diagnostic only: every statement above is about the explicitly displayed model functions, no item of either page lists this example among its dependencies, and no later proof uses it.

6.2A1F1F4step 1.1step 1.2step 1.4step 1.5step 1.6step 4.1step 5.1∎

Audit of hypotheses, degenerate cases and choice. The zero initial derivative $E = 0$ gives the zero Jacobi field at every curvature; it retains model equality and is excluded only from the strict field comparison. The strict monotonicity of step 5.1 is asserted for nondegenerate admissible data only: $θ \in (0, π)$ and the strict triangle inequalities are used through the hypothesis $θ \in (0, π)$ and through $∣ u ∣ < v$ , and the degenerate limits $θ = 0$ and $θ = π$ are excluded, in agreement with the admissibility convention of [F4] and the strict inequalities of [F10]. The boundary case $k = 0$ is treated throughout: the model functions have their $k = 0$ values, the half-angle identities of step 1.6 reduce to the Euclidean cosine law and are proved there directly, the primitive $m$ is $t^{3} / 3$ and the addition identity of step 2.3 is proved separately for $k = 0$ ; the formula of step 5.1 uses only the identity of step 1.6, not a division by $k$ . The case $s > D_{k} / 2$ of the positive-curvature model, where the sine is decreasing in its argument, is handled in step 4.1 through the reflection $t_{s} = D_{k} - s$ ; for $k \leq 0$ no such case occurs since $D_{k} = + \infty$ . The isosceles case $a = b$ , i.e. $u = 0$ , is included: then $h (u) = 0$ and the strict convexity inequality of step 4.1 still separates $h (u)$ from $h (s)$ because $s > 0$ . The endpoint radius and time values of the comparison theorems ( $p$ , the cut locus, $r = 0$ , $t_{0} = π / \sqrt{k}$ ) are excluded exactly as in [F6]–[F9] and are never substituted into a formula; the saturated model volume $V_{k}^{⋆}$ handles $r \geq π / \sqrt{k}$ without evaluating the density at the spherical endpoint. No family of objects is selected: the model geodesics, triangles and Jacobi fields are supplied by the cited model definitions, so the inherited ${A C}_{ω}$ of [A1] suffices and no further choice is used. No converse implication is asserted and no statement of this example is used by any other item; the diagnostic claims of steps 1.5 and 6.1 are consistent with, and independent of, the comparison theorems they test.

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The equality case and the model computations are classical: the model solutions ${sn}_{k}, {cs}_{k}, {ct}_{k}$ and their comparison roles are Datar, Lectures on Riemannian Geometry, §24.1 (printed pp.173–176) and Eschenburg, Comparison Theorems in Riemannian Geometry, §2 (printed pp.6–11); the space-form equalities in the Hessian, Laplacian and volume comparisons are Eschenburg §§3–5 (printed pp.11–20) and Datar §§26–27; the Toponogov comparison with its equality case in constant curvature is Eschenburg §6 (printed pp.21–25) and Lang, Chapter 5, Theorem 5.15 (printed pp.70–71, PDF pp.73–74). The half-angle law of cosines used in step 1.6 is Lang, Riemannian and Metric Geometry, Chapter 5, Lemma 5.1, and the strict monotonicity of the model opposite side in the included angle used there is the same chapter's Lemma 5.2 (printed pp.64–65); the strict monotonicity of the model angle in the curvature proved in step 5.1 is the classical comparison quantity monotonicity that underlies the space-form ordering, and it is derived here from the displayed formulas rather than quoted.

Riemannian Comparison Theorems — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Model jacobi fields in positive zero and negative curvature

Example

Facts & Assumptions

Verification

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Rauch comparison between euclidean and spherical geodesics

Example

Facts & Assumptions

Verification

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Distance hessian and laplacian in space forms

Example

Facts & Assumptions

Proof

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Cartan hadamard for hyperbolic space

Example

Facts & Assumptions

Verification

A flat torus showing simple connectedness is needed for global exp injectivity

Example

Facts & Assumptions

Verification

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Bonnet-Myers for the round sphere

Example

Facts & Assumptions

Verification

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Bishop gromov ratio is constant in the model space

Example

Facts & Assumptions

Verification

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Volume growth in euclidean and hyperbolic space

Example

Facts & Assumptions

Verification

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Toponogov comparison on a round sphere

Example

Facts & Assumptions

Verification

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Positive sectional curvature with no fixed lower bound on a noncompact manifold

Statement refuted

Facts & Assumptions

Counterexample

Ricci lower bound does not control every sectional curvature in dimension at least three

Statement refuted

Facts & Assumptions

Counterexample

Equality cases as diagnostics for all comparison signs

Example

Facts & Assumptions

Verification

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Sources