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Jacobi Fields, Conjugate Points, and the Cut Locus — Examples

1 · Prerequisites

2 · Summary

The examples fix the theory in the model spaces. In Euclidean space the geodesics are the straight lines and the Jacobi fields are the affine fields $A + t B$ , so there are no conjugate points, every straight ray minimizes for all time, and the Hessian of the distance is $(1 / r_{p}) (g - d r_{p} \otimes d r_{p})$ off $p$ . Constant sectional curvature gives the normal modes $\sin (\sqrt{K} t)$ and $\sinh (\sqrt{- K} t)$ : conjugate antipodes on the round sphere at the multiples of $π R$ , conjugate-free geodesics when $K \leq 0$ , explicit index forms, and rotational Killing fields restricting to normal Jacobi fields.

The cut-locus examples compute cut time and the cut locus exactly: the flat circle, the flat rectangular torus from its Dirichlet cell, and the round sphere, whose cut locus of a point collapses to the antipode. Two counterexamples separate the notions: a cut point that is not conjugate, because two minimizing segments arrive there, and an antipodal conjugate point of the round sphere at which infinitely many minimizing meridians meet, so conjugacy does not single out a unique minimizer.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

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Jacobi fields in euclidean space

Example

For $n \geq 0$ , give $R^{n}$ its standard Euclidean metric $g_{E} = \sum_{i = 1}^{n} d x^{i} \otimes d x^{i} .$ Let $I \subseteq R$ be a nondegenerate interval, choose $x, v \in R^{n}$ , and set $γ (t) = x + t v$ . In the standard Cartesian trivialization, a smooth field $J$ along $γ$ is Jacobi if and only if there are constant vectors $A, B \in R^{n}$ such that $J (t) = A + t B (t \in I) .$ This includes the constant geodesic $v = 0$ and the zero-dimensional case $n = 0$ .

Facts & Assumptions

Given: The standard Euclidean metric on $R^{n}$ , a nondegenerate interval $I$ , and $x, v \in R^{n}$ defining $γ (t) = x + t v$ .

[F1]

The standard Euclidean inner product is $⟨ u, w ⟩ = \sum_{i} u_{i} w_{i}$ , so its global Cartesian metric matrix is $δ_{i j}$ ; a smooth symmetric positive-definite coordinate matrix defines a Riemannian metric (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Coordinate criterion for a riemannian metric).

[F2]

In a smooth chart, the coordinate derivations $\partial_{1} ∣_{p}, \dots, \partial_{n} ∣_{p}$ form a basis of $T_{p} M$ (Coordinate derivations form a basis of the tangent space).

[F3]

The connection coefficients are defined by $\nabla_{\partial_{i}} \partial_{j} = \sum_{k} Γ^{k}_{i j} \partial_{k}$ , and the Levi-Civita symbols satisfy $Γ^{k}_{i j} = \frac{1}{2} \sum_{ℓ} g^{k ℓ} (\partial_{i} g_{j ℓ} + \partial_{j} g_{i ℓ} - \partial_{ℓ} g_{i j})$ (Christoffel symbols of an affine connection, Christoffel formula for the levi civita connection).

[F4]

In coordinates, $R^{ℓ}_{k i j} = \partial_{i} Γ^{ℓ}_{j k} - \partial_{j} Γ^{ℓ}_{i k} + Γ^{m}_{j k} Γ^{ℓ}_{i m} - Γ^{m}_{i k} Γ^{ℓ}_{j m}$ (Coordinate formula for the curvature tensor).

[F5]

A smooth curve is geodesic exactly when ${\ddot{x}}^{k} + Γ^{k}_{i j} (x) {\dot{x}}^{i} {\dot{x}}^{j} = 0$ in its coordinates (Coordinate geodesic equation).

[F6]

A smooth vector field along a smooth curve has smooth coefficient functions in a pulled-back frame (Vector field and section along a smooth curve).

[F7]

Along the curve, $D_{t} V = (γ^{*} \nabla)_{\partial / \partial t} V$ and $D_{t} (f V) = f^{'} V + f D_{t} V$ (Covariant derivative along a curve).

[F8]

The Jacobi equation is $D_{t}^{2} J + R (J, \dot{γ}) \dot{γ} = 0$ (Jacobi field).

[F9]

A continuous real function on an order-convex interval whose derivative vanishes at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[F10]

Sums and scalar multiples obey the derivative sum and scalar rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ); from the difference quotient, the identity function has derivative $1$ and a constant function has derivative $0$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[F11]

Every interval is order-convex, and a nondegenerate interval has at least two points (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · Cartesian coordinates and the componentwise Jacobi equation

1.1F1F3

In the global Cartesian chart, [F1] gives the constant metric matrix $g_{i j} = δ_{i j}$ , which is smooth, symmetric, and positive definite. All coordinate derivatives $\partial_{i} g_{j ℓ}$ vanish, so [F3] gives $Γ^{k}_{i j} = 0$ identically.

1.2F3F6F7step 1.1

Write an arbitrary smooth field as $J (t) = \sum_{i} j_{i} (t) \partial_{i} ∣_{γ (t)}$ using [F6]. By the pullback-connection definition in [F7] and the Christoffel definition in [F3], $D_{t} \partial_{i} = \sum_{j} {\dot{γ}}^{j} \nabla_{\partial_{j}} \partial_{i} = \sum_{j, k} {\dot{γ}}^{j} Γ^{k}_{j i} \partial_{k} = 0,$ because step 1.1 gives every $Γ^{k}_{j i} = 0$ . The product rule in [F7], applied twice, gives $D_{t} J = \sum_{i} j_{i}^{'} (t) \partial_{i}$ and $D_{t}^{2} J = \sum_{i} j_{i}^{''} (t) \partial_{i}$ .

2.1F2F4F5F8step 1.1step 1.2

The coordinate functions of $γ$ are $x^{i} + t v^{i}$ , hence ${\ddot{γ}}^{i} = 0$ ; [F5] and step 1.1 show that $γ$ is an affinely parametrized geodesic. Substitution of $Γ = 0$ and its zero derivatives into [F4] gives every curvature component zero; [F2] then gives $R = 0$ . By step 1.2 and [F8], the Jacobi equation is therefore $\sum_{i} j_{i}^{''} \partial_{i} = 0$ , and [F2] implies $j_{i}^{''} = 0$ for every coordinate on the interior of $I$ .

3.1F9F10F11step 2.1

For each $i$ , the function $j_{i}^{'}$ is continuous on $I$ and has derivative $j_{i}^{''} = 0$ at every interior point, so [F9] makes it a constant $B_{i}$ . By [F10], the continuous function $j_{i} (t) - B_{i} t$ has derivative $j_{i}^{'} (t) - B_{i} = 0$ on the interior; another application of [F9] makes it a constant $A_{i}$ . Thus $j_{i} (t) = A_{i} + t B_{i}$ throughout $I$ , including any endpoints by continuity, and the component vectors give $J (t) = A + t B$ .

3.2F7F8step 1.1step 1.2step 2.1

Conversely, for any constant $A, B \in R^{n}$ , the field $J (t) = A + t B$ is smooth. Steps 1.1–1.2 and [F7] give $D_{t} J = B$ in the constant coordinate frame, $D_{t}^{2} J = 0$ , and step 2.1 gives $R = 0$ ; [F8] therefore makes $J$ Jacobi.

4.1F2F6F7F8F11step 1.2step 2.1step 3.1step 3.2∎

If $n = 0$ , the tangent spaces and field contain only zero, represented by the unique $A = B = 0$ ; if $n = 1$ , the same component calculation is scalar. The case $v = 0$ is included because step 2.1 still gives a constant geodesic and step 1.2 allows arbitrary smooth coefficient functions; $A = 0$ , $B = 0$ , or $J = 0$ need no exclusion. At included time endpoints all derivatives are one-sided and the affine formula extends by continuity. Only finitely many coordinates are considered, and the proof makes no choice; singleton intervals are excluded by the nondegeneracy hypothesis. Steps 2.1, 3.1 and 3.2 prove both directions of the stated classification.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 5, “Geodesics of the Model Spaces—Euclidean Space,” printed p.81 / PDF labels P97–98, lines 3393–3399, states that the Euclidean metric has constant coefficients, its Christoffel symbols vanish, and its geodesics are straight lines. Datar, Lectures on Riemannian Geometry, Lecture 22, Example 22.1.1, printed p.160 / PDF label P167, lines 9120–9128, states the component equation $J^{''} = 0$ and the affine solution form but does not derive them; the proof above supplies that calculation. Datar's immediately following statement that every nonzero Jacobi field “has a unique zero” is not used: a nonzero constant affine field has no zero, and in general the valid conclusion is at most one zero.

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Jacobi fields in constant sectional curvature

Example

Assume exactly ${A C}_{ω}$ through the declared constant-curvature interface. Let $(M, g)$ be a finite-dimensional Riemannian manifold without boundary of constant sectional curvature $K \in R$ , let $I \subseteq R$ be a nondegenerate interval containing $0$ , and let $γ : I \to M$ be a unit-speed affinely parametrized geodesic. Put $T = \dot{γ}$ and define

s_{K} (t) = {\begin{cases} \sin (\sqrt{K} t) / \sqrt{K}, & K > 0, \\ t, & K = 0, \\ \sinh (\sqrt{- K} t) / \sqrt{- K}, & K < 0. \end{cases}

Then the normal Jacobi fields $J$ with $J (0) = 0$ are exactly $J (t) = s_{K} (t) E (t),$ where $E$ is a unique parallel normal field along $γ$ . The tangential Jacobi fields are exactly $J (t) = (a t + b) \dot{γ} (t), a, b \in R .$ No completeness is assumed. If an endpoint of $I$ is included, derivatives there are one-sided.

Facts & Assumptions

Given: The constant sectional curvature $K$ , a nondegenerate interval $I$ containing $0$ , and a supplied unit-speed affine geodesic $γ$ on $I$ .

[A1]

${A C}_{ω}$ is countable choice. Its only role here is inherited through the constant-sectional-curvature and curvature-tensor interfaces; the coordinate computations and the unique extensions of supplied vectors use no further choice, and no full AC is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Constant sectional curvature and space form, Curvature tensor of constant sectional curvature).

[F1]

Constant sectional curvature $K$ means that every tangent two-plane has sectional curvature $K$ . In dimensions zero and one this predicate is vacuous for every $K$ (Constant sectional curvature and space form).

[F2]

Under this hypothesis, the curvature operator is $R (X, Y) Z = K (g (Y, Z) X - g (X, Z) Y)$ (Curvature tensor of constant sectional curvature).

[F3]

A Jacobi field is a smooth field satisfying $D_{t}^{2} J + R (J, T) T = 0$ throughout the interval (Jacobi field).

[F4]

The affine geodesic equation gives $D_{t} T = 0$ , and unit speed gives $g (T, T) = 1$ (Geodesic of an affine connection, given).

[F5]

Levi-Civita metric compatibility, expressed in a local frame with metric matrix $H$ and connection matrix $B$ , gives $H^{'} = B^{T} H + H B$ . The along-curve frame formula is $D_{t} (e u) = e (u^{'} + B u)$ , and covariant differentiation also satisfies $D_{t} (f V) = f^{'} V + f D_{t} V$ . Therefore, for fields $U, V$ along $γ$ , $(g (U, V))^{'} = g (D_{t} U, V) + g (U, D_{t} V) .$ (Levi civita connection, Metric compatible connection on a riemannian vector bundle, Local frame formula for covariant differentiation along a curve, Covariant derivative along a curve)

[F6]

Every supplied initial vector along $γ$ has a unique parallel extension to all of $I$ ; a field is parallel when $D_{t} E = 0$ (Existence and uniqueness of parallel sections, Parallel section along a curve).

[F7]

A continuous real function on an interval whose derivative vanishes at every interior point is constant on that interval (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[F8]

Initial value and covariant derivative at $0$ determine a unique Jacobi field along all of $I$ , including when $0$ is an included endpoint (Existence and uniqueness of jacobi fields from initial data).

[F9]

Since $γ$ has unit speed it is nonconstant, and for every smooth $f : I \to R$ , $f T$ is Jacobi exactly when $f (t) = a t + b$ for constants $a, b$ (Tangential jacobi fields are affine multiples of the velocity).

Verification

technique · parallel transport reduces the normal Jacobi equation to a scalar initial-value equation

1.1givenalgebra

Direct differentiation in the three sign cases gives $s_{K}^{''} + K s_{K} = 0$ , $s_{K} (0) = 0$ , and $s_{K}^{'} (0) = 1$ on the full interval, regardless of later zeros of $s_{K}$ .

1.2F9given

Unit speed makes $γ$ nonconstant, so [F9] proves both directions of the tangential classification: a tangential Jacobi field has affine coefficient and every affine coefficient gives a Jacobi field.

2.1F2F3F4F5F6step 1.1

For any parallel normal $E$ , the product rule gives $D_{t} (s_{K} E) = s_{K}^{'} E$ and $D_{t}^{2} (s_{K} E) = s_{K}^{''} E$ ; [F2] and [F4] give $R (s_{K} E, T) T = K (g (T, T) s_{K} E - g (s_{K} E, T) T) = K s_{K} E$ , so [F3] and step 1.1 make it Jacobi, normal, and zero at $0$ .

3.1F4F5F6F7F8step 1.1step 2.1

If $J$ is normal Jacobi with $J (0) = 0$ , differentiating $g (J, T) = 0$ at $0$ using [F5] and [F4] shows $W = D_{t} J (0) ⊥ T (0)$ ; extend $W$ uniquely to a parallel $E$ by [F6]. Then $h = g (E, T)$ has zero derivative by [F5], is zero at $0$ , and so vanishes on $I$ by [F7]. Since [F6] gives $E (0) = W$ , steps 1.1 and 2.1 give $(s_{K} E) (0) = 0$ and $D_{t} (s_{K} E) (0) = s_{K}^{'} (0) E (0) = W$ . Jacobi initial-data uniqueness [F8] now gives $J = s_{K} E$ on all of $I$ , and [F6] makes $E$ unique.

4.1A1F1F3F4F6F8F9step 1.1step 3.1step 1.2∎

A supplied geodesic rules out empty $M$ ; dimension zero has no unit-speed geodesic, and in dimension one the normal subspace is zero, so the normal case is $J = E = 0$ while the tangential result remains valid. [F1] records the vacuous low-dimensional curvature convention. The interval is nondegenerate; included endpoints use one-sided derivatives; $W = 0$ gives $E = J = 0$ ; the zero tangential field has $a = b = 0$ ; and constant geodesics are excluded by unit speed. The three signs of $K$ are covered in step 1.1. Exactly the inherited ${A C}_{ω}$ of [A1] is assumed, and no choice beyond unique extension of a supplied vector is made.

Source notes

Lee, Riemannian Manifolds, Lemma 10.8 and proof, printed pp.179–180 / PDF labels P195–196, lines 6974–7020, derives the normal equation from the constant-curvature tensor formula and solves the scalar equation. His proof uses the dimension of the normal Jacobi space to conclude exhaustiveness; the argument above instead matches the initial derivative and uses the library's Jacobi initial-value uniqueness theorem. Datar, Lectures on Riemannian Geometry, Proposition 24.1.1 and proof, printed pp.174–175 / PDF labels P181–182, lines 9810–9848, gives the same normal-field formula for a complete space form. Completeness is part of Datar's setting and is not used here. The tangential branch is supplied by the separately cited library proposition.

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Conjugate antipodes on the round sphere

Example

Assume exactly ${A C}_{ω}$ through the declared constant-curvature and sphere interfaces. Let $R > 0$ , let $n \geq 2$ , and give $S_{R}^{n} = {x \in R^{n + 1} : ∣ x ∣ = R}$ the round metric induced from Euclidean space. Let $p \in S_{R}^{n}$ , let $v \in T_{p} S_{R}^{n}$ be a unit vector, and let $γ (t) = \exp_{p} (t v), t \in [0, π R],$ be the radial unit-speed geodesic. Then $γ (π R) = - p$ , the antipode of $p$ , and the antipode is conjugate to $p$ along $γ$ with multiplicity $n - 1.$

Facts & Assumptions

Given: The countable-choice axiom ${A C}_{ω}$ ; an integer $n \geq 2$ ; a radius $R > 0$ ; a point $p$ on the round sphere $S_{R}^{n}$ ; and a unit tangent vector $v \in T_{p} S_{R}^{n}$ with radial geodesic $γ (t) = \exp_{p} (t v)$ on $[0, π R]$ .

[A1]

The exact choice assumption is ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ). It is inherited through the induced-sphere constant-curvature interface, the constant-curvature Jacobi classification, and the maximal-geodesic/parallel-extension interfaces; the rank and evaluation computations below make no selection, and no full Axiom of Choice is used.

[F1]

On the radius- $R$ round sphere, for every point $p$ and every unit tangent vector $v$ the cut time is $c_{p} (v) = π R$ (Cut locus of a point on a round sphere, Example).

[F2]

The cut locus of every point of the radius- $R$ round sphere is the singleton containing the antipode: $Cut (p) = {- p}$ (Cut locus of a point on a round sphere, Example).

[F3]

The cut time is the supremum of the positive radial minimizing times of the unit-speed geodesic through $v$ , and the cut locus consists of the cut-time endpoints over all unit directions; the radial endpoint at time $c_{p} (v)$ is therefore a point of $Cut (p)$ (Cut time in a unit tangent direction, Cut point and cut locus of a point).

[F4]

Under ${A C}_{ω}$ , the maximal geodesic with initial velocity $v$ is the curve $t \mapsto \exp_{p} (t v)$ on its interval of definition, so the supplied $γ$ is an affinely parametrized geodesic with $γ (0) = p$ and $\dot{γ} (0) = v$ (Domain and exponential map of a connection, Geodesic of an affine connection).

[F5]

A geodesic of a metric-compatible connection has constant speed; since $∣ \dot{γ} (0) ∣_{g} = ∣ v ∣_{g} = 1$ , the supplied $γ$ has unit speed, so $T : = \dot{γ}$ satisfies $g (T, T) = 1$ and $T (π R) \neq 0$ (Geodesics have constant speed for a metric-compatible connection).

[F6]

For $n \geq 2$ the round sphere $S_{R}^{n}$ with its induced metric has constant sectional curvature $1 / R^{2}$ (The round sphere has positive constant sectional curvature).

[F7]

On a Riemannian manifold of constant sectional curvature $K$ , the curvature operator is $R (X, Y) Z = K (g (Y, Z) X - g (X, Z) Y)$ (Curvature tensor of constant sectional curvature).

[F8]

On a constant-curvature manifold of curvature $K > 0$ , along a unit-speed geodesic, the normal Jacobi fields $J$ with $J (0) = 0$ are exactly $J (t) = s_{K} (t) E (t)$ , where $s_{K} (t) = \sin (\sqrt{K} t) / \sqrt{K}$ and $E$ is a unique parallel normal field; the tangential Jacobi fields are exactly $J (t) = (a t + b) \dot{γ} (t)$ with $a, b \in R$ (Jacobi fields in constant sectional curvature).

[F9]

Metric compatibility of the Levi--Civita connection along a curve gives $(g (U, V))^{'} = g (D_{t} U, V) + g (U, D_{t} V)$ for smooth fields $U, V$ along $γ$ ; a geodesic satisfies $D_{t} \dot{γ} = 0$ (Levi civita connection, Metric compatible connection on a riemannian vector bundle, Local frame formula for covariant differentiation along a curve, Covariant derivative along a curve).

[F10]

The trigonometric special values include $\sin (π / 2) = 1$ and $\sin π = 0$ (Quarter-turn values and shifts by pi/2 and pi).

[F11]

Every prescribed vector at a point has a unique parallel extension to the whole interval, and a field is parallel exactly when $D_{t} E = 0$ (Existence and uniqueness of parallel sections, Parallel section along a curve).

[F12]

For every prescribed initial value and derivative at $0$ there is exactly one Jacobi field along the interval with that data; with $J (0) = 0$ and $D_{t} J (0) = w$ the field is unique (Existence and uniqueness of jacobi fields from initial data).

[F13]

The tangent space $T_{p} M$ of a smooth $n$ -manifold is an $n$ -dimensional real vector space (The tangent space of an n-manifold has dimension n).

[F14]

The Riemannian metric is positive definite, and the map $w \mapsto g_{p} (v, w)$ is a linear functional whose kernel is exactly the orthogonal complement $v^{⊥} = {w \in T_{p} M : g_{p} (v, w) = 0}$ (Riemannian metric and riemannian manifold, Kernel and image of a linear map).

[F15]

For a linear map of a finite-dimensional space, $\dim V = \dim \ker T + \dim im T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F16]

The endpoints $γ (0)$ and $γ (π R)$ are conjugate along $γ$ exactly when the space $K_{γ} (0, π R) = {J \in J (γ) : J (0) = 0, J (π R) = 0}$ contains a nonzero Jacobi field, and then the multiplicity is $\dim_{R} K_{γ} (0, π R)$ (Conjugate points along a geodesic and their multiplicity).

[F17]

A smooth field $J$ along a geodesic is Jacobi exactly when $D_{t}^{2} J + R (J, \dot{γ}) \dot{γ} = 0$ (Jacobi field).

Verification

technique · locate the antipode as the cut-time endpoint, solve the scalar Jacobi equation in constant positive curvature, bound the endpoint-vanishing space by an initial-derivative injection into $v^\perp$, and match it with the $n-1$ normal zero modes

1.1F1F2F3F4F5F9

By [F4] the supplied curve $γ (t) = \exp_{p} (t v)$ is the affinely parametrized geodesic with $γ (0) = p$ and $\dot{γ} (0) = v$ , and by [F5] it is unit speed, so $T : = \dot{γ}$ satisfies $g (T, T) = 1$ , $T (π R) \neq 0$ , and $D_{t} T = 0$ by the geodesic equation in [F9]. By [F1] the cut time of $v$ is $c_{p} (v) = π R$ , so the radial endpoint $γ (π R)$ is the cut-time endpoint; by [F3] it belongs to $Cut (p)$ , and by [F2] $Cut (p) = {- p}$ . Hence $γ (π R) = - p$ , the antipode of $p$ . The interval $[0, π R]$ is nondegenerate because $R > 0$ .

1.2F6F7F8F10

By [F6] and [F7] the curvature is $K = 1 / R^{2}$ , so the constant-curvature scalar solution is $s_{K} (t) = R \sin (t / R)$ . Hence $s_{K} (0) = 0$ , and by [F10], $s_{K} (π R) = R \sin π = 0$ while $s_{K} (π R / 2) = R \sin (π / 2) = R \neq 0$ .

2.1F7F9F17step 1.1

Let $J$ be any Jacobi field along $γ$ and put $φ (t) = g (J (t), T (t))$ . Since $D_{t} T = 0$ , two applications of the product rule [F9] give $φ^{''} = g (D_{t}^{2} J, T) .$ By the Jacobi equation [F17], $D_{t}^{2} J = - R (J, T) T$ , and the constant-curvature formula [F7] gives $R (J, T) T = K (g (T, T) J - g (J, T) T) .$ Pairing with $T$ and using $g (T, T) = 1$ yields $g (R (J, T) T, T) = K (φ - φ) = 0$ , so $φ^{''} = 0$ and $φ (t) = a t + b$ is affine.

2.2F8F9F11F15step 1.2

Conversely, fix $w \in v^{⊥}$ and let $E$ be the unique parallel field along $γ$ with $E (0) = w$ [F11]. Since $(g (E, T))^{'} = g (D_{t} E, T) + g (E, D_{t} T) = 0$ by [F9] and $g (E (0), T (0)) = g (w, v) = 0$ , the field $E$ is normal. The normal classification [F8] therefore makes $J_{w} (t) = s_{K} (t) E (t)$ a Jacobi field with $J_{w} (0) = 0$ , and step 1.2 gives $J_{w} (π R) = s_{K} (π R) E (π R) = 0$ , so $J_{w} \in K_{γ} (0, π R)$ . The assignment $w \mapsto J_{w}$ is linear and injective: if $J_{w} = 0$ , then evaluating at $π R / 2$ gives $s_{K} (π R / 2) E (π R / 2) = 0$ , and since $s_{K} (π R / 2) \neq 0$ , the parallel field $E$ vanishes at one point, hence $E = 0$ by uniqueness in [F11] and $w = E (0) = 0$ . Therefore $\dim K_{γ} (0, π R) \geq \dim v^{⊥}$ by [F15].

3.1F12F15step 1.1step 2.1

Suppose instead that $J \in K_{γ} (0, π R)$ . By step 2.1, $φ = g (J, T)$ is affine, and $φ (0) = g (0, T (0)) = 0$ while $φ (π R) = g (0, T (π R)) = 0$ ; since $π R > 0$ , an affine function with two distinct zeros vanishes identically, so $φ \equiv 0$ . In particular $g (D_{t} J (0), T (0)) = φ^{'} (0) = 0$ , that is $W : = D_{t} J (0) \in v^{⊥}$ . By [F12], $J$ is the unique Jacobi field with $J (0) = 0$ and $D_{t} J (0) = W$ , so the map $J \mapsto D_{t} J (0)$ is an injective linear map from $K_{γ} (0, π R)$ into $v^{⊥}$ , and $\dim K_{γ} (0, π R) \leq \dim v^{⊥}$ by [F15].

4.1F13F14F15F16step 2.2step 3.1

The linear functional $w \mapsto g_{p} (v, w)$ on $T_{p} M$ has kernel $v^{⊥}$ by [F14] and is surjective because $g_{p} (v, v) > 0$ gives $g_{p} (v, λ v) = λ g_{p} (v, v)$ for every $λ \in R$ . Since $\dim T_{p} M = n$ by [F13], rank--nullity [F15] gives $\dim v^{⊥} = n - 1$ . Combining the two bounds of steps 2.2 and 3.1 yields $\dim K_{γ} (0, π R) = n - 1$ . By the conjugacy definition [F16], the antipode $γ (π R) = - p$ is conjugate to $p$ along $γ$ , with multiplicity $n - 1$ .

5.1A1F16step 1.1step 4.1

Boundary and choice audit. The sphere is nonempty and $p$ is supplied, so no empty case arises; $n \geq 2$ excludes the zero- and one-dimensional spheres, where the constant-curvature normal classification [F8] would not apply with positive curvature; the parameter interval $[0, π R]$ is nondegenerate and $γ$ is a nonconstant unit-speed geodesic, so no constant-geodesic or degenerate-segment case occurs. The zero Jacobi field is excluded from witnessing conjugacy by [F16], and the endpoint-vanishing space is computed exactly, not merely bounded. Exactly the inherited ${A C}_{ω}$ of [A1] is assumed. This example asserts conjugacy at the antipode and its multiplicity; it makes no if-and-only-if claim about geodesics other than the exhibited radial one.

$□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed pp.179–183 / PDF labels P195–P199, Lemma 10.8 solves the constant-curvature normal Jacobi equation and the surrounding discussion records that antipodal points of the round sphere are conjugate. Datar, Lectures on Riemannian Geometry, Proposition 24.1.1, printed pp.174–175 / PDF labels P181–182, gives the same normal-field formula for space forms, and Lecture 22 §22.3 gives the endpoint-vanishing definition of multiplicity used here. The transport of the endpoint argument to the radius- $R$ sphere, the cut-time identification of the antipode, and the rank computation of the normal space are proved locally above from the declared interfaces.

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No conjugate points in nonpositive constant curvature

Example

Assume exactly ${A C}_{ω}$ through the declared dependencies. Let $(M, g)$ be a finite-dimensional Riemannian manifold without boundary of constant sectional curvature $K \leq 0$ , let $I \subseteq R$ be an interval with nonempty interior, and let $γ : I \to M$ be a nonconstant affinely parametrized geodesic. Then for all $a < b$ in $I$ the points $γ (a)$ and $γ (b)$ are not conjugate along $γ ∣_{[a, b]}$ : the real vector space of Jacobi fields along $γ ∣_{[a, b]}$ vanishing at both endpoints is ${0}$ . So no pair of distinct times of $γ$ is joined by a vanishing Jacobi field, and no positive multiplicity occurs along $γ$ .

Facts & Assumptions

Given: The constant-curvature manifold $(M, g)$ with $K \leq 0$ , the interval $I$ , the nonconstant affinely parametrized geodesic $γ$ , and a nondegenerate subinterval $[a, b] \subseteq I$ with a Jacobi field $J$ along $γ$ satisfying $J (a) = J (b) = 0$ .

[A1]

Countable choice is the assumption ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ). It is inherited through the curvature and Riemann tensor interfaces (Curvature tensor of constant sectional curvature, Algebraic symmetries of the Riemann tensor). The direct computation below uses no further choice, and no full Axiom of Choice is assumed.

[F1]

Conjugacy along a segment and vanishing spaces: $γ (a)$ and $γ (b)$ are conjugate along $γ ∣_{[a, b]}$ exactly when some nonzero Jacobi field along the segment vanishes at both endpoints, and the multiplicity is the dimension of that space; for a constant geodesic the space is ${0}$ (Conjugate points along a geodesic and their multiplicity).

[F2]

A Jacobi field satisfies $D_{t}^{2} J + R (J, T) T = 0$ on the interval, with one-sided derivatives at an included endpoint (Jacobi field, Covariant derivative along a curve).

[F3]

An affinely parametrized geodesic satisfies $D_{t} T = 0$ (Geodesic of an affine connection). For a geodesic of a metric-compatible connection the quantity $g (T, T) = ∣ \dot{γ} ∣^{2}$ and the speed are constant on $I$ (Geodesics have constant speed for a metric-compatible connection).

[F4]

Levi-Civita metric compatibility, expressed in a local frame with metric matrix $H$ and connection matrix $B$ , gives $H^{'} = B^{T} H + H B$ ; the along-curve frame formula is $D_{t} (e u) = e (u^{'} + B u)$ . Hence for fields $U, V$ along $γ$ the product rule $(g (U, V))^{'} = g (D_{t} U, V) + g (U, D_{t} V)$ holds (Levi civita connection, Metric compatible connection on a riemannian vector bundle, Local frame formula for covariant differentiation along a curve, Covariant derivative along a curve).

[F5]

Field quantities are continuous up to included endpoints: a smooth field along the closed segment and the smooth metric give continuous functions $t \mapsto g (U (t), V (t))$ there, with the one-sided endpoint convention of Covariant derivative along a curve, and $g$ is a smooth symmetric bilinear positive-definite tensor, so $∣ U ∣^{2} = g (U, U) \geq 0$ with equality only for $U = 0$ (Riemannian metric and riemannian manifold).

[F6]

Under constant sectional curvature $K$ the curvature operator is $R (X, Y) Z = K (g (Y, Z) X - g (X, Z) Y)$ (Curvature tensor of constant sectional curvature).

[F7]

The Riemann tensor is skew in its last two slots: $Rm (X, Y, Z, W) = - Rm (X, Y, W, Z)$ , so $Rm (J, T, T, T) = 0$ (Algebraic symmetries of the Riemann tensor).

[F8]

A continuous function on an interval whose derivative vanishes at every interior point is constant; consequently two continuous functions with equal interior derivatives differ by a constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[F9]

A function twice differentiable on an open interval with nonnegative second derivative there is convex, and convexity means the convex-combination inequality for all weights in $[0, 1]$ (A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative, Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval).

Verification

Proof technique: show that every endpoint-vanishing Jacobi field is normal, then apply convexity to the squared norm of a normal Jacobi field.

1.1

For every Jacobi field $J$ along $γ$ the function $φ = g (J, T)$ is affine on $[a, b]$ . [F2, F3, F4, F7, F8] By [F3] and [F4], $φ^{'} = g (D_{t} J, T) + g (J, D_{t} T) = g (D_{t} J, T)$ , and then $φ^{''} = g (D_{t}^{2} J, T) + g (D_{t} J, D_{t} T) = g (D_{t}^{2} J, T)$ . By the Jacobi equation [F2] and last-pair skewness [F7], $φ^{''} = - g (R (J, T) T, T) = - Rm (J, T, T, T) = 0$ at every interior point. Since $φ^{'}$ is continuous on $[a, b]$ with vanishing derivative inside, [F8] makes $φ^{'}$ constant, and a further application of [F8] makes $φ$ affine: $φ (t) = α t + β$ .

1.2

Let $J$ be a normal Jacobi field along $γ$ , so $g (J, T) = 0$ on $[a, b]$ , and put $u (t) = ∣ J (t) ∣^{2}$ . Then $D_{t}^{2} J = - K c^{2} J$ and $u^{''} = - 2 K c^{2} u + 2 ∣ D_{t} J ∣^{2}$ on $(a, b)$ , where $c^{2} = g (T, T)$ is the constant squared speed. [F2, F3, F4, F5, F6] The curvature term of the Jacobi equation is $R (J, T) T = K (g (T, T) J - g (J, T) T) = K c^{2} J$ by [F6] and normality, so [F2] gives $D_{t}^{2} J = - K c^{2} J$ . The product rule [F4] gives $u^{'} = 2 g (D_{t} J, J)$ and then $u^{''} = 2 g (D_{t}^{2} J, J) + 2 g (D_{t} J, D_{t} J) = - 2 K c^{2} u + 2 ∣ D_{t} J ∣^{2}$ , where $c^{2} = g (T, T)$ is constant by [F3] and both $u$ and $∣ D_{t} J ∣^{2}$ are continuous up to the endpoints by [F5].

2.1

Every Jacobi field $J$ with $J (a) = J (b) = 0$ is normal on $[a, b]$ . [given, step 1.1] By step 1.1, $φ = g (J, T)$ is affine; the endpoint hypotheses give $φ (a) = g (0, T (a)) = 0$ and $φ (b) = g (0, T (b)) = 0$ . An affine function vanishing at the two distinct points $a < b$ is identically zero: writing $φ (t) = α t + β$ , the two equations give $α (b - a) = 0$ , hence $α = β = 0$ . Therefore $g (J, T) = 0$ throughout $[a, b]$ .

2.2

For a normal Jacobi field $J$ along $γ$ with $J (a) = J (b) = 0$ , the function $u = ∣ J ∣^{2}$ satisfies $u^{''} \geq 0$ on $(a, b)$ because $K \leq 0$ ; hence $u$ is convex on $(a, b)$ by [F9], and $u \geq 0$ on $[a, b]$ with $u (a) = u (b) = 0$ and $u$ continuous on $[a, b]$ . [F5, F9, given, step 1.2] By step 1.2, $u^{''} = - 2 K c^{2} u + 2 ∣ D_{t} J ∣^{2}$ ; since $K \leq 0$ and $u \geq 0$ the first term is nonnegative, and the second is nonnegative as well, so $u^{''} \geq 0$ on $(a, b)$ . [F9] therefore makes $u$ convex on the open interval. Nonnegativity and the endpoint values come from [F5] and the hypotheses, and continuity up to the endpoints is [F5].

3.1

If $J$ is a Jacobi field along $γ$ with $J (a) = J (b) = 0$ , then $J$ vanishes identically on $[a, b]$ . [F5, step 2.1, step 2.2] By step 2.1 such a $J$ is normal, so step 2.2 applies to $u = ∣ J ∣^{2}$ . Fix $t \in (a, b)$ . For $a < t_{1} < t < t_{2} < b$ convexity gives $u (t) \leq λ u (t_{1}) + (1 - λ) u (t_{2})$ with $λ = (t_{2} - t) / (t_{2} - t_{1}) \in (0, 1)$ . Letting $t_{1} ↓ a$ and $t_{2} ↑ b$ and using the endpoint values and continuity gives $u (t) \leq 0$ ; with $u (t) \geq 0$ this forces $u (t) = 0$ , and positive definiteness of $g$ gives $J (t) = 0$ . As $t \in (a, b)$ was arbitrary, $J \equiv 0$ on $[a, b]$ , the endpoint values being given.

4.1

Consequently $γ (a)$ and $γ (b)$ are not conjugate along $γ ∣_{[a, b]}$ and no positive multiplicity occurs. [F1, given, step 3.1] By step 3.1 the only Jacobi field along the segment vanishing at both endpoints is the zero field, so the vanishing space is ${0}$ ; by [F1] the endpoints are not conjugate along the segment, and the multiplicity, being defined only for a conjugate pair, does not arise. Since $a < b$ were arbitrary in $I$ , no pair of distinct times of $γ$ is conjugate along the corresponding subsegment.

4.2

Boundary, degeneracy and choice audit. [A1, F1, F2, F5, step 1.1, step 2.1, step 2.2, step 3.1] The subinterval $[a, b]$ is nondegenerate by hypothesis, and included endpoints carry the one-sided conventions of [F2] and [F5]. If $M$ is empty there is no geodesic; in dimension zero the only field is zero, so the vanishing space is ${0}$ by [F1]; in dimension one a field normal to $T$ vanishes, so step 2.1 already forces $J = 0$ . The constant-geodesic case is excluded by the hypothesis that $γ$ is nonconstant and would in any case be a non-conjugate case by the explicit clause of [F1]. The zero field is the only endpoint-vanishing Jacobi field, which is exactly the claim, and the case $K = 0$ is included in the estimate $u^{''} \geq 0$ ; the argument never divides by $K$ . The function $u$ is a squared length and is allowed to vanish on all of $[a, b]$ ; no strict convexity is asserted. Assumption [A1] is inherited from the curvature and Riemann-tensor suppliers, and no selection or countable family is used. The example proves the stated one-way non-conjugacy claim and asserts no converse. $□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed pp.173–190, develops Jacobi fields and conjugate points and treats the constant-curvature model solutions; Datar, Lectures on Riemannian Geometry, §23.3 (printed pp.165–169) and Lecture 24, Proposition 24.1.1 (printed pp.174–175), gives the space-form Jacobi classification; Eschenburg, Comparison Theorems in Riemannian Geometry, §2 (PDF labels P4–P8), records the model solutions in constant curvature. The convexity argument for the squared norm of a normal Jacobi field and the affineness of $g (J, \dot{γ})$ are carried out above rather than quoted.

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Killing jacobi fields from rotations

Example

Let $n \geq 1$ and let $S^{n} = {x \in R^{n + 1} : ⟨ x, x ⟩ = 1}$ carry the round metric induced from the Euclidean inner product. Let $A$ be the skew-symmetric linear map that rotates the first two coordinates, $A e_{1} = e_{2}, A e_{2} = - e_{1}, A e_{j} = 0 (j \geq 3),$ let $R_{θ} = \exp (θ A)$ be the rotation by the angle $θ$ in the first two coordinates, and let $X (x) = A x$ be its generating tangent field on $S^{n}$ . Then $X$ is a Killing field, and for every affinely parametrized geodesic $γ$ of $S^{n}$ the restriction $t \mapsto X (γ (t))$ is a Jacobi field along $γ$ . In particular, the rotational field restricts to a Jacobi field along every great-circle geodesic of the round sphere.

Facts & Assumptions

Given: The integer $n \geq 1$ , the unit round sphere $S^{n}$ with its induced metric, the rotation generator $A$ and the rotation matrices $R_{θ}$ , and the field $X (x) = A x$ .

[F1]

The unit sphere $S^{n}$ is a regular level set of $x \mapsto ⟨ x, x ⟩$ , hence a smooth boundaryless $n$ -manifold; at each $x \in S^{n}$ its tangent space is the kernel $x^{⊥}$ of the derivative, the inclusion is an immersion, and the restricted Euclidean inner product is the round Riemannian metric (A regular level set is an embedded submanifold, The tangent space of a regular level set is the kernel, Pullback of a riemannian metric is riemannian exactly for immersions, Riemannian metric and riemannian manifold, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F2]

On the first two coordinates the map $A$ is the matrix $(\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ and it annihilates the remaining coordinates, so $A^{T} = - A$ , $A^{2}$ is the negative of the orthogonal projection onto $span {e_{1}, e_{2}}$ , and $A^{3} = - A$ . Consequently $R_{θ} = I + \sin θ A + (1 - \cos θ) A^{2}, R_{θ}^{T} = R_{- θ}, R_{θ + φ} = R_{θ} R_{φ},$ the last because $A$ commutes with $A^{2}$ and the trigonometric addition formulas hold; moreover $\frac{d}{d θ} R_{θ} = A R_{θ} = R_{θ} A,$ since $A^{3} = - A$ gives $A R_{θ} = A + \sin θ A^{2} - (1 - \cos θ) A = \cos θ A + \sin θ A^{2}$ (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine, algebra).

[F3]

Each $R_{θ}$ preserves the Euclidean inner product: $⟨ R_{θ} u, R_{θ} v ⟩ = ⟨ u, v ⟩$ for all $u, v \in R^{n + 1}$ , because $R_{θ}^{T} R_{θ} = R_{- θ} R_{θ} = R_{0} = I$ by [F2]. In particular $∣ R_{θ} x ∣ = ∣ x ∣$ , so $R_{θ}$ maps $S^{n}$ to itself (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Parity and the Pythagorean identity for sine and cosine).

[F4]

The curve $θ \mapsto R_{θ} x$ is the maximal integral curve of the field $X$ through $x$ , so the local flow of $X$ is $Φ (θ, x) = R_{θ} x$ (Local and global flows generated by a vector field, The derivatives of sine and cosine are cosine and minus sine).

[F5]

A smooth local diffeomorphism $F$ of a Riemannian manifold is a (local) isometry when $F^{*} g = g$ ; for the sphere this means $g_{F (x)} (d F_{x} u, d F_{x} v) = g_{x} (u, v)$ for tangent vectors $u, v$ (Riemannian isometry and local isometry).

[F6]

The Lie derivative of the metric along a vector field with flow $Φ$ is $L_{X} g = \frac{d}{d θ} ∣_{θ = 0} Φ_{θ}^{*} g$ , and a tensor field is flow-invariant exactly when its Lie derivative vanishes (The Lie derivative of a tensor field, A tensor field is flow-invariant exactly when its Lie derivative vanishes).

[F7]

For a smooth vector field $X$ with $L_{X} g = 0$ (a Killing field in the sense of that proposition) and an affinely parametrized geodesic $γ$ , the restriction $J (t) = X (γ (t))$ is a Jacobi field along $γ$ (Killing fields restrict to Jacobi fields along geodesics, Geodesic of an affine connection).

[F8]

Every constant-speed parametrization of a great circle of the round sphere is a geodesic (Great circles as round-sphere geodesics).

Verification

technique · exhibit the rotation flow, check it preserves the round metric by orthogonality of the rotation matrices, and invoke the Killing-field proposition

1.1F1F2F3

By [F1] the unit sphere is a boundaryless Riemannian $n$ -manifold whose round metric is the ambient inner product on tangent vectors. The map $X (x) = A x$ is linear, hence smooth, and it is tangent to the sphere: differentiating $∣ R_{θ} x ∣^{2} = 1$ in $θ$ at $0$ , or directly $⟨ A x, x ⟩ = x^{T} A^{T} x = - x^{T} A x = - ⟨ A x, x ⟩$ , gives $X (x) \in T_{x} S^{n} = x^{⊥}$ .

1.2F2F3F4

By [F2] the explicit matrices satisfy $R_{θ}^{T} = R_{- θ}$ , $R_{θ + φ} = R_{θ} R_{φ}$ , and $\frac{d}{d θ} R_{θ} = A R_{θ}$ . Hence by [F3] each $R_{θ}$ is orthogonal and preserves $S^{n}$ , and the curve $θ \mapsto R_{θ} x$ is defined for all real $θ$ with derivative $\frac{d}{d θ} R_{θ} x = A R_{θ} x = X (R_{θ} x)$ and value $x$ at $θ = 0$ .

2.1F1F3F5step 1.2

The flow $Φ (θ, x) = R_{θ} x$ of step 1.2 consists of isometries of the round sphere: for $x \in S^{n}$ and $u, v \in T_{x} S^{n}$ , the pushforward is $d (R_{θ})_{x} u = R_{θ} u$ and $g_{R_{θ} x} (R_{θ} u, R_{θ} v) = ⟨ R_{θ} u, R_{θ} v ⟩ = ⟨ u, v ⟩ = g_{x} (u, v)$ by [F1] and [F3]. So $Φ_{θ}^{*} g = g$ for every $θ$ .

3.1F6step 2.1

By [F6] the Lie derivative of the round metric along $X$ is $L_{X} g = \frac{d}{d θ} ∣_{θ = 0} Φ_{θ}^{*} g$ , and by step 2.1 every term equals $g$ , so $L_{X} g = 0$ : the rotational field $X$ is a Killing field.

4.1F7F8step 3.1

Now let $γ$ be an affinely parametrized geodesic of the round sphere. By [F7], applied to the Killing field $X$ of step 3.1, the restricted field $J (t) = X (γ (t))$ is a Jacobi field along $γ$ . By [F8] every great-circle geodesic is among these affinely parametrized geodesics, and the rotational field restricts to a Jacobi field along it.

5.1F1F7step 1.1step 4.1

Boundary and choice audit. The sphere is nonempty for every $n \geq 1$ ; the rotation angle is a real parameter and the first two coordinates exist because $n \geq 1$ , so $X$ is nonzero; the flow is global, so no local-domain restriction or completeness hypothesis enters. No choice principle is used: the generator $A$ , the matrices $R_{θ}$ , and the field $X$ are all supplied explicitly, and the cited suppliers are choice-free. The example asserts the Jacobi property for every affinely parametrized geodesic; it makes no if-and-only-if claim.

$□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapters 8 and 10, treats the round sphere as the basic example of a space of constant curvature and includes the rotational isometries; Datar, Lectures on Riemannian Geometry, Lectures 15, 22 and 24, discusses Killing fields and their Jacobi restrictions. The coordinate computation of the rotation flow and the verification that the round metric is invariant under it are carried out above rather than quoted.

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Cut locus of a point on a round sphere

Example

Assume exactly ${A C}_{ω}$ through the declared dependencies. Let $S_{R}^{n} = {x \in R^{n + 1} : ⟨ x, x ⟩ = R^{2}}, R > 0, n \geq 2,$ with the Riemannian metric induced by the Euclidean inner product. For every $p \in S_{R}^{n}$ and unit $v \in T_{p} S_{R}^{n}$ , the cut time is $c_{p} (v) = π R,$ and the cut locus is the singleton $Cut (p) = {- p} .$

Facts & Assumptions

Given: $R > 0$ , an integer $n \geq 2$ , and a point $p$ on the radius- $R$ round sphere, with the metric induced from $R^{n + 1}$ .

[A1]

${A C}_{ω}$ is the countable-choice assumption of The Axiom of Countable Choice ( ${A C}_{ω}$ ). It is used through the induced Levi-Civita connection, maximal-geodesic uniqueness, Hopf--Rinow, and the cut-time/cut-locus interfaces below. No full axiom of choice is used.

[F1]

The level set $S_{R}^{n} = F^{- 1} (R^{2})$ for $F (x) = ⟨ x, x ⟩$ is nonempty and regular: $R e_{n + 1} \in S_{R}^{n}$ , and at every $x \in S_{R}^{n}$ , $d F_{x} (x) = 2 R^{2} \neq 0$ . Thus A regular level set is an embedded submanifold gives a smooth boundaryless $n$ -manifold, while The tangent space of a regular level set is the kernel gives $T_{x} S_{R}^{n} = x^{⊥}$ . The inclusion is an immersion, so Pullback of a riemannian metric is riemannian exactly for immersions and Riemannian metric and riemannian manifold make the restricted Euclidean inner product a Riemannian metric (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F2]

The radial scaling $x \mapsto x / R$ is a homeomorphism from $S_{R}^{n}$ to the unit sphere $S^{n}$ . Since $n \geq 2$ , For $n \geq 2$ , the sphere $S^{n - 1}$ is path-connected and connected makes $S^{n}$ , and hence $S_{R}^{n}$ , connected. Nonemptiness and the boundaryless manifold property were checked in [F1].

[F3]

In Euclidean coordinates the ambient Levi-Civita derivative is ordinary differentiation (The euclidean levi civita connection). For an embedded submanifold, the induced connection is the tangential projection of the ambient derivative (Induced connection and second fundamental form); under [A1], The induced connection is Levi–Civita identifies it with the Levi-Civita connection of the induced metric.

[F4]

An affine geodesic satisfies $D_{t} \dot{γ} = 0$ (Geodesic of an affine connection). Under [A1], every initial tangent vector has a unique maximal geodesic (Existence uniqueness and smooth dependence of geodesics), and Geodesically complete Riemannian manifold means each such maximal domain is $R$ .

[F5]

Under [A1], Hopf--Rinow says that a nonempty, connected, boundaryless Riemannian manifold is metrically complete exactly when it is geodesically complete (Hopf–Rinow theorem).

[F6]

On this connected manifold, distance is the infimum of lengths of piecewise- $C^{1}$ joining paths, and length is the sum of the integrals of Riemannian speed over their smooth pieces (Riemannian distance on a connected manifold, Riemannian speed and length).

[F7]

Euclidean Cauchy--Schwarz holds (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs); the chain rule applies (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ); sine and cosine have their usual derivatives and satisfy $\sin^{2} s + \cos^{2} s = 1$ (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine). Also $\cos π = - 1$ and $\sin π = 0$ (Quarter-turn values and shifts by pi/2 and pi). The principal inverse cosine is the continuous inverse of cosine restricted to $[0, π]$ and has range $[0, π]$ (Principal inverse sine and inverse cosine); on $(- 1, 1)$ its derivative is $- 1 / \sqrt{1 - y^{2}}$ (For $- 1 < y < 1$ , $(\arcsin y)^{'} = 1 / \sqrt{1 - y^{2}}$ and $(\arccos y)^{'} = - 1 / \sqrt{1 - y^{2}}$ ).

[F8]

For a differentiable real-valued function with integrable derivative, the vector-valued fundamental theorem and norm inequality give $∣ f (b) - f (a) ∣ \leq \int_{a}^{b} ∣ f^{'} ∣$ (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable). Pointwise comparison of integrable functions passes to their integrals (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[F9]

Under the completeness, connectedness, boundaryless, and ${A C}_{ω}$ hypotheses, the cut time is the supremum of the positive radial minimizing times and the cut locus consists of finite cut-time endpoints (Cut time in a unit tangent direction, Cut point and cut locus of a point).

Verification

technique · explicit great-circle geodesics and an angular length bound

1.1A1F1F2given

The defining function $F (x) = ⟨ x, x ⟩$ has derivative $d F_{x} (w) = 2 ⟨ x, w ⟩$ . Since $x \neq 0$ on $S_{R}^{n}$ , this derivative is surjective onto $R$ at every level-set point. By [F1], $S_{R}^{n}$ is a nonempty, connected, boundaryless Riemannian $n$ -manifold and $T_{x} S_{R}^{n} = x^{⊥}$ .

1.2F7algebra

Fix $q \in S_{R}^{n}$ and any piecewise- $C^{1}$ path $α : [0, 1] \to S_{R}^{n}$ from $p$ to $q$ . On each smooth piece put $u (s) = ⟨ p, α (s) ⟩ / R^{2}$ . Cauchy--Schwarz in [F7] gives $∣ u ∣ \leq 1$ . Differentiating $∣ α ∣^{2} = R^{2}$ gives $⟨ α, \dot{α} ⟩ = 0$ , and hence $u^{'} = \frac{⟨ p - u α, \dot{α} ⟩}{R^{2}}, ∣ p - u α ∣ = R \sqrt{1 - u^{2}}, ∣ u^{'} ∣ \leq \frac{\sqrt{1 - u^{2}}}{R} ∣ \dot{α} ∣ .$ For $0 < ε < 1$ , define $θ_{ε} = \arccos ((1 - ε) u)$ . Its argument lies strictly between $- 1$ and $1$ , so [F7] and the chain rule give $∣ θ_{ε}^{'} ∣ = \frac{(1 - ε) ∣ u^{'} ∣}{\sqrt{1 - (1 - ε)^{2} u^{2}}} \leq \frac{(1 - ε) \sqrt{1 - u^{2}}}{R \sqrt{1 - (1 - ε)^{2} u^{2}}} ∣ \dot{α} ∣ \leq \frac{∣ \dot{α} ∣}{R} .$ The last inequality follows because $1 - (1 - ε)^{2} u^{2} - (1 - ε)^{2} (1 - u^{2}) = 1 - (1 - ε)^{2} \geq 0$ .

2.1A1F3F4F7step 1.1

Fix $x \in S_{R}^{n}$ and $w \in T_{x} S_{R}^{n}$ . If $w = 0$ , the constant curve is a geodesic on all of $R$ . Otherwise set $a = ∣ w ∣$ and define $γ_{x, w} (t) = \cos (a t / R) x + \frac{R}{a} \sin (a t / R) w, t \in R .$ Since $⟨ x, w ⟩ = 0$ and $∣ x ∣ = R$ , $∣ w ∣ = a$ , the trigonometric identity in [F7] gives $∣ γ_{x, w} (t) ∣ = R$ ; differentiating gives $γ_{x, w} (0) = x$ , ${\dot{γ}}_{x, w} (0) = w$ , $∣ {\dot{γ}}_{x, w} (t) ∣ = a$ , and ${\ddot{γ}}_{x, w} (t) = - (a^{2} / R^{2}) γ_{x, w} (t)$ . The acceleration is normal to the sphere, so its tangential projection vanishes. By [F3] this curve is an affinely parametrized geodesic. It is defined for every real $t$ ; uniqueness in [F4] identifies it with the maximal geodesic for $(x, w)$ . Therefore the sphere is geodesically complete.

2.2F6F7F8step 1.2algebra

On each smooth piece, [F8] yields $∣ θ_{ε} (b) - θ_{ε} (a) ∣ \leq \int_{a}^{b} ∣ θ_{ε}^{'} ∣ \leq R^{- 1} \int_{a}^{b} ∣ \dot{α} ∣$ . Summing over the pieces and using the triangle inequality gives $∣ θ_{ε} (1) - θ_{ε} (0) ∣ \leq L_{g} (α) / R$ . As $ε ↓ 0$ , continuity of the principal inverse cosine in [F7] gives $R \arccos ⁣ (\frac{⟨ p, q ⟩}{R^{2}}) \leq L_{g} (α) .$ This lower bound holds for every competitor in [F6].

3.1A1F5F9step 1.1step 2.1

The nonempty, connected, boundaryless hypotheses follow from step 1.1. Hopf--Rinow [F5] and geodesic completeness from step 2.1 therefore make $(S_{R}^{n}, d_{g})$ metrically complete, as required by [F9].

3.2F1F6F7step 2.1step 2.2

Put $θ = \arccos (⟨ p, q ⟩ / R^{2}) \in [0, π]$ . Since $∣ p ∣ = ∣ q ∣ = R$ and $⟨ p, q ⟩ = R^{2} \cos θ$ , $∣ q - \cos θ p ∣^{2} = R^{2} \sin^{2} θ$ , so $θ = 0$ implies $q = p$ and $θ = π$ implies $q = - p$ . If $0 < θ < π$ , define $v = (q - \cos θ p) / (R \sin θ)$ . Direct inner-product calculation gives $v \in p^{⊥}$ and $∣ v ∣ = 1$ . The geodesic in step 2.1 with initial data $(p, v)$ reaches $q$ at time $R θ$ and has unit speed. If $θ = 0$ then $q = p$ and the constant path has length zero. If $θ = π$ then $q = - p$ ; because $n \geq 2$ , $p^{⊥}$ contains a unit vector $v$ , and the same formula reaches $- p$ at time $π R$ with unit speed. Thus a path of length $R θ$ exists in every case. Combining this upper bound with step 2.2 proves $d_{g} (p, q) = R \arccos ⁣ (\frac{⟨ p, q ⟩}{R^{2}}) .$

4.1A1F7F9step 2.1step 3.2

Let $v \in T_{p} S_{R}^{n}$ be any unit vector. Step 2.1 gives its radial geodesic $γ_{v} (t) = \cos (t / R) p + R \sin (t / R) v$ . Hence $⟨ p, γ_{v} (t) ⟩ / R^{2} = \cos (t / R)$ , and step 3.2 gives $d_{g} (p, γ_{v} (t)) = R \arccos (\cos (t / R)) .$ For $0 < t \leq π R$ , the inverse-cosine definition in [F7] makes this equal to $t$ . For $t > π R$ , its range $[0, π]$ gives $d_{g} (p, γ_{v} (t)) \leq π R < t$ . Thus the positive minimizing-time set is exactly $(0, π R]$ , including the endpoint, and [F9] yields $c_{p} (v) = π R$ .

5.1A1F1F3F4F5F9step 2.1step 3.2step 4.1∎

At the finite endpoint, every unit direction has $γ_{v} (π R) = - p$ . Since $n \geq 2$ , the tangent space has unit vectors, so the cut-point set in [F9] is nonempty and equals ${- p}$ . The empty case cannot occur because $p$ is supplied and $S_{R}^{n}$ contains $R e_{n + 1}$ ; the zero- and one-dimensional spheres are outside the stated $n \geq 2$ claim. The zero initial vector in the completeness check was treated in step 2.1, and the degenerate angular case $q = p$ in step 3.2. The radius is strictly positive; the cut-time definition excludes $t = 0$ , includes $t = π R$ , and every later time fails strictly by step 4.1. Choosing a unit vector for the single antipodal path in step 3.2 uses only the nonzero finite-dimensional tangent space for that supplied $p$ , not a choice function on a family. Exactly the declared ${A C}_{ω}$ is propagated through [F3]--[F5] and [F9]; no full AC is used. This example makes no iff claim.

Source locator

Eschenburg, Comparison Theorems in Riemannian Geometry, Section 5, Example 5.1 (printed p.16), states without proof that every unit-sphere direction has cut time $π$ and remains shortest up to the antipode. Lee, Riemannian Manifolds, Chapter 10, printed p.190 / PDF P206, lines 7559--7571, supplies the finite cut-point and cut-locus convention. The induced-sphere geodesic formula, angular distance lower bound, radius- $R$ scaling, and exact cut-time calculation are proved locally above.

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Cut locus of a point on a flat circle

Example

Assume ${A C}_{ω}$ . For $L > 0$ , let $C_{L} = R / (L Z)$ have the flat metric whose period-coordinate charts carry $d x^{2}$ , so its circumference is $L$ . For every $p = [x] \in C_{L}$ , the two unit tangent directions are $v_{+} = \partial_{x}$ and $v_{-} = - \partial_{x}$ . Both have cut time $c_{p} (v_{+}) = c_{p} (v_{-}) = \frac{L}{2},$ and $Cut (p) = {[x + L / 2]},$ the antipode of $p$ . At the cut point the two opposite semicircles are distinct minimizing geodesics.

Facts & Assumptions

Given: A positive circumference $L$ , the quotient circle $C_{L}$ , its flat metric, and a base point $p = [x]$ .

[A1]

${A C}_{ω}$ means every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )). It is assumed only through the geodesic maximality, Hopf--Rinow, and cut-time/cut-locus interfaces below.

[F1]

The library uses the quotient circle $R / Z$ as the flat circle factor with local metric $d θ^{2}$ (Hopf–Rinow on a flat cylinder). Rescaling by $x = L θ$ gives $C_{L} = R / (L Z)$ ; its period-coordinate transitions are $x \mapsto x + k L$ , so $d x^{2}$ is a well-defined smooth positive metric. These charts make $C_{L}$ a one-dimensional manifold without boundary. The class $[0]$ makes it nonempty, and projected straight segments connect any two classes.

[F2]

In the quotient model, $[r] = [s]$ exactly when their difference is an integer period (The circle as $S^{1} = R / Z$ with basepoint $[0]$ after rescaling).

[F3]

The metric coefficient in a period chart is $1$ , so $∣ u \partial_{x} ∣ = ∣ u ∣$ (Riemannian metric and riemannian manifold, Pointwise norm and angle from a riemannian metric). Riemannian speed is the norm of velocity and length is its integral over the smooth pieces (Riemannian speed and length); distance on this connected manifold is the infimum of lengths (Riemannian distance on a connected manifold).

[F4]

Every interval shorter than one period embeds as an open quotient arc (The quotient map is open, and every interval shorter than one embeds in $R / Z$ ). Thus the quotient projection is a covering: these arcs are homeomorphic to their interval lifts and have disjoint period translates as full preimage (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings). Every path has a unique lift from a specified starting point (Existence and uniqueness of path lifts through a covering map). The local inverse charts make a lift of a piecewise $C^{1}$ path piecewise $C^{1}$ and preserve its coordinate speed.

[F5]

On each closed smooth piece, the vector-valued fundamental theorem gives $f (b) - f (a) = \int_{a}^{b} f^{'}$ , and the norm of an integral is at most the integral of the norm (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

[F6]

For every real $z$ there is an integer $n$ with $n \leq z < n + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[F7]

In coordinates, constant metric coefficients give zero Christoffel symbols, and a geodesic satisfies ${\ddot{x}}^{k} + Γ^{k}_{i j} {\dot{x}}^{i} {\dot{x}}^{j} = 0$ (Christoffel formula for the levi civita connection, Coordinate geodesic equation). Under ${A C}_{ω}$ , initial data determine a unique maximal geodesic (Existence uniqueness and smooth dependence of geodesics); geodesic completeness means that every such maximal domain is $R$ (Geodesically complete Riemannian manifold).

[F8]

Under ${A C}_{ω}$ , Hopf--Rinow says a nonempty connected boundaryless Riemannian manifold is metrically complete exactly when it is geodesically complete (Hopf–Rinow theorem).

[F9]

Cut time is the supremum of the positive times $t$ for which the radial distance equals $t$ , and the cut locus consists of the finite cut-time endpoints over all unit directions; both definitions assume completeness, connectedness, no boundary, and ${A C}_{ω}$ (Cut time in a unit tangent direction, Cut point and cut locus of a point).

Proof

technique · quotient lift and nearest-period calculation

1.1F1F2F3given

Write $q_{L} (x) = [x]$ . In every period chart the metric coefficient is $1$ , and chart changes are translations by $k L$ , so the metric is well-defined. The explicit curve $t \mapsto [x + t (y - x)]$ connects any two classes, and $[0]$ shows the circle is nonempty. These facts and [F1]-[F2] establish the quotient metric model; [F3] gives its tangent norms. Thus $C_{L}$ is nonempty, connected, one-dimensional, and boundaryless.

1.2F2F3F4F5

Fix $p = [x]$ and $q = [y]$ . Let $α : [0, 1] \to C_{L}$ be any piecewise $C^{1}$ path from $p$ to $q$ , and lift it from $x$ using [F4]. Its endpoint is $y + k L$ for some $k \in Z$ , by the quotient equivalence in [F2]. Choose a finite subdivision $0 = a_{0} < \dots < a_{m} = 1$ on which $α$ is $C^{1}$ on each piece; the local inverse charts for the covering make $\tilde{α}$ $C^{1}$ on those same pieces. Put $Δ_{i} = \tilde{α} (a_{i + 1}) - \tilde{α} (a_{i})$ . On each piece the lift has the same speed as $α$ , and [F5] gives $Δ_{i} = \int_{a_{i}}^{a_{i + 1}} {\tilde{α}}^{'} (t) d t$ and $∣ Δ_{i} ∣ \leq \int_{a_{i}}^{a_{i + 1}} ∣ {\tilde{α}}^{'} (t) ∣ d t$ . Hence the triangle inequality and the length definition give $∣ y + k L - x ∣ = ∣ \tilde{α} (1) - \tilde{α} (0) ∣ = ∣ \sum_{i = 0}^{m - 1} Δ_{i} ∣ \leq \sum_{i = 0}^{m - 1} ∣ Δ_{i} ∣ \leq \sum_{i = 0}^{m - 1} \int_{a_{i}}^{a_{i + 1}} ∣ {\tilde{α}}^{'} (t) ∣ d t = L_{g} (α) .$ Thus $L_{g} (α) \geq ∣ y + k L - x ∣ \geq \inf_{j \in Z} ∣ y + j L - x ∣$ . Taking the infimum over $α$ gives $d_{g} (p, q) \geq \inf_{j} ∣ y + j L - x ∣$ .

2.1A1F7step 1.1

For any point $[x]$ and any tangent vector $u \partial_{x}$ , define $γ (t) = [x + u t]$ for all $t \in R$ . In each period chart its coordinate is affine, the metric coefficients are constant, and [F7] gives $Γ = 0$ and the geodesic equation. Thus this is a global geodesic with the prescribed initial data. Uniqueness of the maximal geodesic in [F7] implies that every maximal geodesic is defined on all of $R$ . Hence $C_{L}$ is geodesically complete.

2.2F2F3F6step 1.2

Put $z = (y - x) / L$ and choose $n = ⌊ z + 1 / 2 ⌋$ by [F6]. Then $- 1 / 2 \leq z - n < 1 / 2$ . For every integer $j$ , this makes $n$ a nearest integer to $z$ : if $j \geq n + 1$ then $j - z > 1 / 2$ , and if $j \leq n - 1$ then $z - j \geq 1 / 2$ . Therefore the infimum in step 1.2 is attained and $\inf_{j \in Z} ∣ y + j L - x ∣ = ∣ y - n L - x ∣ \leq L / 2.$ For each integer $j$ , the projected straight segment $σ_{j} (t) = [x + t (y + j L - x)]$ , $0 \leq t \leq 1$ , joins $p$ to $q$ and has constant speed $∣ y + j L - x ∣$ . Choosing $j = - n$ attains the lower bound in step 1.2, so $d_{g} ([x], [y]) = \min_{j \in Z} ∣ y + j L - x ∣ .$ This also proves $d_{g} ([x], [y]) \leq L / 2$ for all $p, q$ .

3.1A1F8F9step 1.1step 2.1

The nonempty, connected, boundaryless hypotheses were checked in step 1.1, and step 2.1 gives geodesic completeness. Hopf--Rinow [F8] therefore makes $(C_{L}, d_{g})$ complete, as required by [F9].

3.2A1F3F9step 2.1step 2.2

By [F3], the unit tangent vectors at $p$ are exactly $v_{+} = \partial_{x}$ and $v_{-} = - \partial_{x}$ . Their radial geodesics are $γ_{\pm} (t) = [x \pm t]$ by step 2.1. For $0 < t < L / 2$ , step 2.2 gives $d_{g} (p, γ_{\pm} (t)) = t$ . At $t = L / 2$ , both adjacent period representatives have absolute displacement $L / 2$ , so the same equality holds and there are two distinct minimizing semicircle segments: $s \mapsto [x + s L / 2]$ and $s \mapsto [x - s L / 2]$ , $0 \leq s \leq 1$ . They have the same length $L / 2$ and different interior points. If $t > L / 2$ , step 2.2 gives $d_{g} (p, γ_{\pm} (t)) \leq L / 2 < t$ . Thus the positive minimizing-time set in each direction is exactly $(0, L / 2]$ , and [F9] gives $c_{p} (v_{+}) = c_{p} (v_{-}) = L / 2$ .

4.1A1F2F3F6F7F9step 1.1step 3.1step 3.2∎

The two finite cut endpoints coincide: $γ_{+} (L / 2) = [x + L / 2] = [x - L / 2] = γ_{-} (L / 2)$ . The quotient relation makes this point independent of the representative $x$ ; it is the antipode. Since [F3] gives exactly the two unit directions and [F9] defines the cut locus as their finite cut endpoints, $Cut (p) = {[x + L / 2]}$ . The computation holds for every $p$ . The circle is nonempty and one-dimensional, so the empty- and zero-dimensional cases are not instances; in dimension one both unit directions were handled. The degenerate value $t = 0$ is excluded from the cut-time supremum, the included endpoint $L / 2$ still minimizes, and every later time fails strictly. The only choice assumption is the declared ${A C}_{ω}$ used through [F7]-[F9]; the quotient lifts are unique from the specified start and the nearest period is fixed by [F6], with no appeal to full AC. There is no iff claim.

Source locator

Lee, Riemannian Manifolds, Chapter 10, printed p.190 / PDF P206, lines 7559–7571, defines cut points and cut loci and notes that on the flat cylinder geodesics wrapping more than halfway are not minimizing. That passage does not give the circle quotient-distance calculation; steps 1.1–4.1 prove it here.

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Cut locus on a flat rectangular torus from the Dirichlet cell

Example

Assume exactly the declared ${A C}_{ω}$ assumption. Let $a, b > 0$ , let $Λ = a Z \times b Z$ , and give $Q = R^{2} / Λ$ its quotient flat metric. Write $q : R^{2} \to Q$ for the quotient projection and fix $p = [x_{0}] \in Q$ . The period-coordinate charts identify $T_{p} Q$ with $R^{2}$ . Put $D = [- a / 2, a / 2] \times [- b / 2, b / 2], D^{\circ} = (- a / 2, a / 2) \times (- b / 2, b / 2) .$ Define the radial tangent cut domain including zero by $C_{p} = {0} \cup {t u : ∣ u ∣ = 1, 0 < t < c_{p} (u)} .$ Then $C_{p} = D^{\circ}$ ; equivalently, the positive tangent cut domain is $D^{\circ} ∖ {0}$ . For each unit $u = (u_{1}, u_{2})$ , $c_{p} (u) = \min ⁣ ({\frac{a}{2 ∣ u_{1} ∣} : u_{1} \neq 0} \cup {\frac{b}{2 ∣ u_{2} ∣} : u_{2} \neq 0}),$ where the displayed minimum is over the nonempty set of defined terms. Moreover, $Cut (p) = {[x_{0} + w] : w \in \partial D} .$ Opposite edges of $D$ are identified in $Q$ . A relative-interior edge class has exactly two nearest lattice lifts from $p$ , and the four corners represent one class with four nearest lattice lifts.

Facts & Assumptions

Given: Positive periods $a, b$ , the lattice quotient set $Q = R^{2} / Λ$ with its quotient topology, the quotient map $q$ , and $p = [x_{0}]$ . The period-coordinate flat metric is constructed below.

[A1]

Exactly ${A C}_{ω}$ is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Its uses below are through geodesic existence and uniqueness, Hopf--Rinow, and the cut-time and cut-locus interfaces.

[F1]

In the quotient topology, $V \subseteq Q$ is open exactly when $q^{- 1} [V]$ is open in $R^{2}$ ; the quotient classes are those of the given lattice equivalence relation (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[F2]

For every real $z$ there is a unique integer $n$ with $n \leq z < n + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[F3]

A topological $2$ -manifold without boundary is Hausdorff, second-countable, and locally homeomorphic to open subsets of $R^{2}$ ; a smooth manifold has a maximal smooth atlas (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces, Smooth manifolds and their smooth charts).

[F4]

A Riemannian metric is a smooth positive-definite symmetric two-tensor; in coordinates it is enough to check that its matrices are smooth, symmetric, and positive definite, with the usual tensor change-of-coordinate law (Riemannian metric and riemannian manifold, Coordinate criterion for a riemannian metric).

[F5]

The Euclidean inner product on $R^{2}$ induces the norm $∣ z ∣ = \sqrt{⟨ z, z ⟩}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F6]

Squaring preserves order on nonnegative reals, and every nonnegative real has its unique nonnegative square root, so coordinatewise minima minimize the Euclidean norm (Squaring is monotone on the nonnegatives, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[F7]

A chart tangent vector has the pointwise Riemannian norm determined by its metric matrix (Pointwise norm and angle from a riemannian metric).

[F8]

The Euclidean norm satisfies the triangle inequality (The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).

[F9]

A covering has evenly covered neighbourhoods, and every path has a unique lift after its starting point is fixed (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings, Existence and uniqueness of path lifts through a covering map). A piecewise $C^{1}$ path has a finite chartwise subdivision; once this quotient is shown to be covered, composing those pieces with local inverse charts makes its lift piecewise $C^{1}$ (Piecewise c one curve on a manifold).

[F10]

Riemannian speed is the norm of velocity, length is the sum of its speed integrals over the finite smooth pieces, and length is unchanged by finite subdivision. On a connected Riemannian manifold distance is the infimum of these lengths (Riemannian speed and length, Riemannian length is independent of piecewise c one subdivision, Riemannian distance on a connected manifold).

[F11]

For a piecewise $C^{1}$ curve in $R^{2}$ , its continuous coordinate derivatives are integrable componentwise, and the vector-valued fundamental theorem gives the displacement as the integral of the derivative; the norm of that integral is at most the integral of the speed. Finite sums telescope (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz, For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable, A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Laws of finite sums and finite products).

[F12]

In coordinates the Levi-Civita symbols are given by the Christoffel formula, and geodesics satisfy the coordinate geodesic equation (Christoffel formula for the levi civita connection, Coordinate geodesic equation). Under [A1], initial data determine a unique maximal geodesic; the exponential map evaluates it at time one, and geodesic completeness means all maximal geodesics have domain $R$ (Existence uniqueness and smooth dependence of geodesics, Domain and exponential map of a connection, Geodesically complete Riemannian manifold).

[F13]

A path-connected space is connected (Every path-connected space is connected, and every path component lies inside a component).

[F14]

Under [A1], Hopf--Rinow makes a geodesically complete nonempty connected boundaryless Riemannian manifold complete for its Riemannian distance (Hopf–Rinow theorem).

[F15]

For a complete, connected, boundaryless Riemannian manifold, cut time in a unit direction is the supremum of positive times at which radial distance equals time (Cut time in a unit tangent direction).

[F16]

The cut locus consists of the finite cut-time endpoints over all unit directions (Cut point and cut locus of a point).

Proof

technique · quotient path lifting and the rectangular Dirichlet cell

1.1F2F5F6

For $z \in R$ , set $n = ⌊ z + 1 / 2 ⌋$ using [F2]. Then $- 1 / 2 \leq z - n < 1 / 2$ . If $m \geq n + 1$ , then $∣ z - m ∣ \geq ∣ z - n ∣$ ; if $m \leq n - 1$ , then again $∣ z - m ∣ \geq ∣ z - n ∣$ . Equality for a second integer occurs exactly at the half-period boundary. Applying this to $z = (y_{i} - x_{i}) / L_{i}$ for $L_{1} = a, L_{2} = b$ shows that each coordinate has an attained nearest lattice translate. The squared Euclidean norm is minimized coordinatewise, so $\min_{λ \in Λ} ∣ y - x + λ ∣ = \sqrt{δ_{1}^{2} + δ_{2}^{2}}, δ_{i} = \min_{k \in Z} ∣ y_{i} - x_{i} + k L_{i} ∣ .$ This minimum is zero exactly when $y - x \in Λ$ .

2.1F1F3F4F13step 1.1

The quotient projection is open: for open $U \subseteq R^{2}$ , $q^{- 1} (q (U)) = ⋃_{λ \in Λ} (U + λ),$ which is open, so [F1] makes $q (U)$ open. The images of rational balls therefore form a countable base. If $q (x) \neq q (y)$ , [F2] and step 1.1 give $δ = \min_{λ \in Λ} ∣ y - x + λ ∣ > 0$ . Choose $r > 0$ with $2 r < δ$ . If $q (B (x, r))$ met $q (B (y, r))$ , some $x^{'} \in B (x, r)$ and $y^{'} \in B (y, r)$ would satisfy $x^{'} - y^{'} \in Λ$ , yielding a lattice translate of $y - x$ of norm $∣ y^{'} - x^{'} ∣ < 2 r$ , a contradiction. Thus $Q$ is Hausdorff. For $0 < ε < \min (a, b) / 2$ , each rectangle $U_{x} = x + (- ε, ε)^{2}$ has pairwise disjoint lattice translates; $q ∣_{U_{x}}$ is an open continuous bijection onto the open set $q (U_{x})$ , hence a chart. On overlaps the coordinate changes are locally translations by elements of $Λ$ , so they are smooth. The coordinate metric matrices are the constant identity matrix; [F4] makes this a smooth positive flat metric. The same disjoint-translate description of $q^{- 1} (q (U_{x}))$ shows these charts evenly cover their images, so $q$ is a covering and a local isometry. Projected straight segments join every pair of classes, so $Q$ is path-connected and connected by [F13]. It is nonempty because it contains $q (0)$ , and its charts have no boundary.

3.1F5F7F8F9F10F11step 1.1step 2.1

Fix $x, y \in R^{2}$ and any piecewise $C^{1}$ path $α$ in $Q$ from $q (x)$ to $q (y)$ . Lift it from $x$ by [F9]. On each path piece lying in a quotient chart, its lift lies in one translated sheet and is the chartwise inverse of $α$ ; hence the lift is piecewise $C^{1}$ and has the same coordinate speed. Its endpoint is $y + λ$ for some $λ \in Λ$ . Refine to a common finite subdivision and write $Δ_{j} = \tilde{α} (t_{j + 1}) - \tilde{α} (t_{j})$ . By [F11], $∣ Δ_{j} ∣ \leq \int_{t_{j}}^{t_{j + 1}} ∣ {\tilde{α}}^{'} (t) ∣ d t .$ The increments telescope, the triangle inequality in [F8] bounds their sum, and the local isometry preserves speed. Therefore $∣ y - x + λ ∣ = ∣ \tilde{α} (1) - \tilde{α} (0) ∣ \leq \sum_{j} ∣ Δ_{j} ∣ \leq L_{g} (α) .$ By step 1.1 this is at least $\min_{μ \in Λ} ∣ y - x + μ ∣$ . Taking the infimum over all paths gives the same lower bound for $d_{g} (q (x), q (y))$ .

3.2A1F12step 2.1

For every $x \in R^{2}$ and $w \in T_{q (x)} Q ≅ R^{2}$ , define $γ_{x, w} (t) = q (x + t w)$ for all $t \in R$ . In every quotient chart its coordinates are affine and the metric coefficients are constant, so [F12] gives zero Christoffel symbols and the geodesic equation. This is a global geodesic with the prescribed initial data. Uniqueness in [F12] shows that every maximal geodesic is this one; hence $Q$ is geodesically complete and $\exp_{q (x)} (w) = q (x + w)$ . This includes $w = 0$ , whose geodesic is constant.

4.1F10step 1.1step 3.1

Step 1.1 supplies a translate $λ_{*}$ attaining the minimum. The projection of the straight segment from $x$ to $y + λ_{*}$ has constant speed $∣ y - x + λ_{*} ∣$ , so [F10] gives its length as that norm. Combining it with step 3.1 proves the attained quotient-distance formula $d_{g} (q (x), q (y)) = \min_{λ \in Λ} ∣ y - x + λ ∣ = \sqrt{δ_{1}^{2} + δ_{2}^{2}} .$

4.2A1F14F15F16step 2.1step 3.2

The model is nonempty, connected, boundaryless, and geodesically complete by steps 2.1 and 3.2. Hopf--Rinow [F14], under exactly [A1], makes $(Q, d_{g})$ complete. Thus the completeness hypotheses of [F15] and [F16] hold.

5.1A1F5F7F15step 1.1step 3.2step 4.1

Let $u = (u_{1}, u_{2})$ be a unit vector and put $τ (u) = \min ⁣ ({\frac{a}{2 ∣ u_{1} ∣} : u_{1} \neq 0} \cup {\frac{b}{2 ∣ u_{2} ∣} : u_{2} \neq 0}) .$ The set is nonempty and $0 < τ (u) < \infty$ . For $0 < t \leq τ (u)$ , $t u \in D$ ; step 1.1 says zero is a nearest lattice translate, including a tie when a coordinate reaches a face. Steps 3.2 and 4.1 then give $d_{g} (p, \exp_{p} (t u)) = ∣ t u ∣ = t$ . If $t > τ (u)$ , choose a nonzero coordinate $u_{i}$ attaining the minimum. Then $∣ t u_{i} ∣ > L_{i} / 2$ , where $L_{1} = a, L_{2} = b$ . Adding the opposite period to that coordinate strictly reduces its absolute value, leaves the other coordinate unchanged, and gives a projected straight competitor of length strictly less than $∣ t u ∣ = t$ . Hence $d_{g} (p, \exp_{p} (t u)) < t$ . The minimizing-time set is exactly $(0, τ (u)]$ , so [F15] gives $c_{p} (u) = τ (u)$ .

6.1F15F16step 2.1step 4.2step 5.1

For $v \neq 0$ , write $v = ∣ v ∣ u$ with $∣ u ∣ = 1$ . The formula for $τ (u)$ in step 5.1 shows $∣ v ∣ < τ (u)$ exactly when $∣ v_{1} ∣ < a / 2$ and $∣ v_{2} ∣ < b / 2$ . Adjoining $v = 0$ proves both inclusions $C_{p} = D^{\circ}$ ; omitting zero gives the positive domain $D^{\circ} ∖ {0}$ . For each unit $u$ , step 5.1 puts $τ (u) u$ on $\partial D$ , so every finite cut endpoint lies in ${[x_{0} + w] : w \in \partial D}$ . Conversely, given $w \in \partial D$ , $w \neq 0$ ; put $u = w / ∣ w ∣$ . For each nonzero coordinate, its candidate exit time is $L_{i} ∣ w ∣ / (2 ∣ w_{i} ∣) \geq ∣ w ∣$ , with equality on every face coordinate, so $τ (u) = ∣ w ∣$ and $[x_{0} + w]$ is a cut endpoint. Thus $Cut (p) = {[x_{0} + w] : w \in \partial D}$ , proving both set inclusions.

7.1F2F5step 1.1step 6.1

In one coordinate, a point strictly between the two half-periods has one nearest period representative, while either endpoint $\pm L_{i} / 2$ has exactly two, differing by $L_{i}$ . Independence of the two coordinates makes each relative-interior edge point have exactly two nearest lattice lifts and each corner have four. Opposite edges differ by a lattice vector and therefore have the same image. A vertical-edge image and a horizontal-edge image meet only when both coordinates are half-periods; all four corners then give the same corner class. This is the stated branch intersection.

8.1

The torus is nonempty and two-dimensional; $a, b > 0$ rule out a collapsed period. The zero tangent vector is included in $C_{p}$ but is not a unit direction. Directions with one zero coordinate are covered by omitting that coordinate's quotient from the minimum in step 5.1. The cut-time supremum is over $t > 0$ , its endpoint $τ (u)$ is included and minimizing, and every later time fails strictly. The two set equalities in step 6.1 were proved in both directions. The only choice assumption is [A1] through geodesic uniqueness, Hopf--Rinow [F14] and the cut interfaces [F15, F16]; coordinate rounding is determined by [F2], path lifts are unique from a specified start, and no full AC or family selection is used. No other dimension or iff claim is made. [A1, F2, F12, F14, F15, F16, step 2.1, step 3.2, step 4.2, step 5.1, step 6.1, step 7.1] QED

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed p.190 / PDF P206, lines 7559–7568, defines cut points and the cut locus and gives the flat-cylinder example where geodesics wrapping past halfway cease to minimize. This passage does not establish the rectangular-torus distance formula or Dirichlet cell; those are derived locally in steps 1.1–4.2.

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A cut point that is not conjugate because two minimizers arrive

Statement refuted

Refuted claim. Let $(M, g)$ be a complete, connected Riemannian manifold without boundary, let $p \in M$ , and let $q \in Cut (p)$ be a cut point of $p$ reached by a minimizing geodesic segment $γ$ from $p$ to $q$ . Then $p$ and $q$ are conjugate along $γ$ .

Facts & Assumptions

Given: The countable-choice axiom ${A C}_{ω}$ ; a circumference $L > 0$ ; the flat circle $C_{L} = R / (L Z)$ with period-coordinate metric $d x^{2}$ ; the base point $p = [x]$ ; the antipode $q = [x + L / 2]$ ; and the two opposite semicircles $γ_{+} (t) = [x + t]$ and $γ_{-} (t) = [x - t]$ on $[0, L / 2]$ .

[A1]

The exact choice assumption is ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ). It is inherited only through the flat-circle example's maximal-geodesic, Hopf--Rinow and cut-time interfaces; the one-dimensional Jacobi calculation below uses no choice, and no full Axiom of Choice is used.

[F1]

In the flat circle $C_{L}$ the antipodal cut-time statement holds: both unit directions have cut time $L / 2$ , the cut locus is the single antipode $Cut (p) = {[x + L / 2]}$ , and at that cut point the two opposite semicircles are distinct minimizing geodesics (Cut locus of a point on a flat circle).

[F2]

The points $γ (a)$ and $γ (b)$ are conjugate along the affinely parametrized geodesic segment $γ : [a, b] \to M$ exactly when the space $K_{γ} (a, b) = {J \in J (γ) : J (a) = 0, J (b) = 0}$ of Jacobi fields vanishing at both endpoints contains a nonzero field (Conjugate points along a geodesic and their multiplicity).

[F3]

A smooth field $J$ along $γ$ is Jacobi exactly when it satisfies the Jacobi equation $D_{t}^{2} J + R (J, \dot{γ}) \dot{γ} = 0$ (Jacobi field).

[F4]

In a coordinate chart of a Riemannian metric, the Levi-Civita Christoffel symbols are given by the Christoffel formula; for the period chart of $C_{L}$ the metric matrix is the constant $1 \times 1$ matrix $(1)$ , so the formula gives $Γ^{1}_{11} = 0$ (Christoffel formula for the levi civita connection).

[F5]

In a coordinate chart the curvature components are recovered from the Christoffel symbols by the coordinate curvature formula; when the Christoffel symbols vanish identically in the chart, every component $R^{ℓ}_{k i j}$ vanishes, so the curvature operator $R$ is zero at each point of the chart (Coordinate formula for the curvature tensor).

[F6]

For a local frame $e$ with connection form $B (t) = ω_{γ (t)} (\dot{γ} (t))$ , a field $V (t) = e (γ (t)) v (t)$ has covariant derivative $D_{t} V = e (γ (t)) (v^{'} (t) + B (t) v (t))$ (Local frame formula for covariant differentiation along a curve); a section is parallel exactly when $D_{t} V = 0$ (Parallel section along a curve). In the period chart of $C_{L}$ the coordinate field $\partial_{x}$ is a frame along either semicircle, and its connection form vanishes because it is built from the symbols $Γ^{1}_{11} = 0$ of [F4].

Proof

technique · realize the two minimizing semicircles of the flat circle example, note that the flat connection makes the coordinate frame parallel and the curvature zero, and solve the resulting scalar Jacobi equation $j''=0$ with two endpoint zeros

1.1F1

By [F1], $q$ lies in $Cut (p)$ , and $γ_{+}$ and $γ_{-}$ are distinct minimizing geodesics from $p$ to $q$ with common length $L / 2$ . In a period chart containing the image of the closed semicircle, $γ_{+}$ lifts to the affine line $s \mapsto x + s$ and $γ_{-}$ lifts to $s \mapsto x - s$ ; each lift has constant unit coordinate speed on $[0, L / 2]$ .

1.2F4F5F6

In that period chart the metric coefficient is the constant $1$ , so [F4] gives vanishing Christoffel symbols, [F5] gives $R = 0$ at every point of the chart, and [F6] makes the coordinate field $\partial_{x}$ a parallel frame along the semicircle, since its connection form is built from those vanishing symbols.

2.1F3F5F6step 1.2

Let $J$ be a smooth vector field along $γ_{+}$ . On the chart domain write $J (t) = j (t) \partial_{x}$ for a smooth real function $j$ . As $\partial_{x}$ is parallel by step 1.2, the frame formula [F6] gives $D_{t} J = j^{'} \partial_{x}$ and $D_{t}^{2} J = j^{''} \partial_{x}$ , while the curvature term vanishes because $R = 0$ by step 1.2. Thus, by [F3], $J$ is a Jacobi field along $γ_{+}$ exactly when the scalar equation $j^{''} (t) = 0$ holds on $[0, L / 2]$ , whose solutions are $j (t) = a + b t$ with constants $a, b \in R$ .

3.1F3step 2.1algebra

Suppose $J$ is a Jacobi field along $γ_{+}$ with $J (0) = J (L / 2) = 0$ . By step 2.1, $j (t) = a + b t$ with $a = j (0) = 0$ and $a + b L / 2 = j (L / 2) = 0$ . Since $L / 2 > 0$ , the second equation forces $b = 0$ , hence $j = 0$ and $J = 0$ . The same calculation applies to $γ_{-}$ , whose chart lift $s \mapsto x - s$ has the same constant metric coefficient and the same vanishing symbols.

4.1F2step 3.1

Consequently $K_{γ_{+}} (0, L / 2) = {0}$ and $K_{γ_{-}} (0, L / 2) = {0}$ : the only Jacobi field along either minimizing semicircle vanishing at both endpoints is the zero field. By the conjugacy definition [F2], $p$ and $q$ are not conjugate along either semicircle.

5.1A1F1F2step 1.1step 4.1

Therefore $q$ is a cut point of $p$ , reached by the two distinct minimizing geodesics $γ_{+}$ and $γ_{-}$ , and it is not conjugate to $p$ along either of them; the claim in the Statement refuted is false. Boundary and choice audit: the two semicircles have positive length $L / 2$ , so both endpoint times are distinct and the endpoint zeros are genuine; the circle is one-dimensional and nonempty, with no degenerate interval and no constant geodesic among the two witnesses; the zero field is the only endpoint-vanishing Jacobi field and it cannot witness conjugacy by [F2]. Exactly ${A C}_{ω}$ is used, and only to invoke the flat-circle example's cut-time interface by [A1]; the Jacobi computation selects nothing. The refuted claim is a one-way implication, so no converse case arises.

$□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed p.190 / PDF label P206, lines 7559–7571, observes that the flat cylinder has no conjugate points and that wrappings longer than halfway stop minimizing; that geometry is realized here on the flat circle, and the local nonconjugacy calculation is carried out rather than quoted. Datar, Lectures on Riemannian Geometry, Appendix C.14.2, printed p.278 / PDF labels P284–285, lines 14021–14050, states Klingenberg's lemma with an exercise outline relating nonconjugate cut points to two minimizing geodesics; the explicit circle witness above does not use that exercise as a proof.

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A conjugate point at which there are many geodesics

Statement refuted

Refuted claim. Let $(M, g)$ be a complete, connected Riemannian manifold without boundary, and let $γ : [a, b] \to M$ be a minimizing geodesic segment from $p = γ (a)$ to $q = γ (b)$ such that $p$ and $q$ are conjugate along $γ$ . Then $γ$ is the unique minimizing geodesic segment from $p$ to $q$ .

Assume ${A C}_{ω}$ . The claim is false. On the round sphere $S_{R}^{n}$ of radius $R > 0$ with $n \geq 2$ , let $p \in S_{R}^{n}$ and let $q = - p$ be the antipode of $p$ . For every unit $v \in T_{p} S_{R}^{n}$ the radial geodesic $γ_{v} (t) = \exp_{p} (t v), 0 \leq t \leq π R,$ is a minimizing geodesic from $p$ to $q$ along which $q$ is conjugate to $p$ with multiplicity $n - 1$ , and the unit directions produce infinitely many pairwise distinct such meridians. So at the conjugate point $q$ the minimizing geodesic is far from unique: no single meridian is the unique minimizing geodesic from $p$ to $q$ .

Facts & Assumptions

Given: The countable-choice axiom ${A C}_{ω}$ ; a radius $R > 0$ ; an integer $n \geq 2$ ; the round sphere $S_{R}^{n}$ with the Riemannian metric $g$ induced by the Euclidean inner product; a point $p \in S_{R}^{n}$ ; and the index set $N$ for the family constructed below.

[A1]

The choice assumption is ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ). It is inherited only through the two sphere examples below (their Hopf--Rinow, cut-time and Jacobi interfaces). The orthonormal pair and the family of directions are built from one finite list and explicit formulas, so no selection from a family is made and no full Axiom of Choice is used.

[F1]

For every unit $v \in T_{p} S_{R}^{n}$ the cut time is $c_{p} (v) = π R$ and the cut locus is the singleton $Cut (p) = {- p}$ (Cut locus of a point on a round sphere, Example).

[F2]

When the cut time is finite, the cut point of $p$ along $γ_{v}$ is $γ_{v} (c_{p} (v))$ , and it is the last minimizing point on that ray: $c_{p} (v) \in A_{p} (v) = {t \geq 0 : d_{g} (p, γ_{v} (t)) = t}$ . In particular $d_{g} (p, γ_{v} (π R)) = π R$ and the radial segment up to the cut time is minimizing (Cut point and cut locus of a point, Definition; Minimizing along a geodesic is an initial interval property, Statement).

[F3]

For every $v \in T_{p} M$ and every $t$ with $t v$ in the domain of $\exp_{p}$ one has $\exp_{p} (t v) = γ_{p, v} (t)$ , the maximal geodesic with $γ_{p, v} (0) = p$ and ${\dot{γ}}_{p, v} (0) = v$ (The exponential map scales geodesic time, Statement; Geodesic of an affine connection, Definition).

[F4]

A geodesic of the Levi-Civita connection has constant speed; for a unit $v$ the radial geodesic $γ_{v}$ therefore has speed $1$ , and its length over $[0, π R]$ is the integral of the constant function $1$ on a smooth piece, namely $π R$ (Geodesics have constant speed for a metric-compatible connection, Statement; Riemannian speed and length, Definition; Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative, Statement). The Riemannian distance is the infimum of lengths of joining curves (Riemannian distance on a connected manifold, Definition), so a curve whose length equals the distance between its endpoints is minimizing. Riemannian metric and riemannian manifold supplies the metric; Levi civita connection supplies metric compatibility of the Levi-Civita connection.

[F5]

For every unit $v \in T_{p} S_{R}^{n}$ the endpoint $γ_{v} (π R) = - p$ is conjugate to $p$ along $γ_{v}$ with multiplicity $n - 1$ (Conjugate antipodes on the round sphere, Example).

[F6]

The sphere $S_{R}^{n}$ is a smooth $n$ -manifold (Smooth manifolds and their smooth charts, Definition). In a chart at $p$ the coordinate derivations $\partial_{1} ∣_{p}, \dots, \partial_{n} ∣_{p}$ form a basis of the tangent space $T_{p} S_{R}^{n}$ (Coordinate derivations form a basis of the tangent space, Statement), where the tangent space is the space of derivations at $p$ (Derivations at a point and the tangent space, Definition).

[F7]

Applying Gram--Schmidt to the linearly independent list of the first two basis vectors of [F6] gives orthonormal vectors $e_{1}, e_{2} \in T_{p} S_{R}^{n}$ with $g_{p} (e_{i}, e_{j}) = δ_{i j}$ for $i, j \in {1, 2}$ (Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans, Statement).

[F8]

On $T_{p} S_{R}^{n}$ the round metric is the restriction of the Euclidean inner product (Cut locus of a point on a round sphere, Example): a bilinear, symmetric, positive-definite real inner product (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Real and complex inner product spaces, with the inner product linear in the first argument, The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product), so $g_{p}$ is linear in each argument and $∥ w ∥_{g} = \sqrt{g_{p} (w, w)}$ for $w \in T_{p} S_{R}^{n}$ .

[F9]

For every $m \in N$ the embedded natural number satisfies $m \geq 0$ in $R$ , hence $m^{2} \geq 0$ and $1 + m^{2} > 0$ (The natural numbers $N$ (von Neumann), Canonical naturals are positive and strictly increasing). For $a \geq 0$ the nonnegative square root satisfies $(\sqrt{a})^{2} = a$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Statement), and for $a, b \geq 0$ one has $a \leq b ⟺ a^{2} \leq b^{2}$ (Squaring is monotone on the nonnegatives, Statement).

[F10]

$N$ is not equinumerous with any natural number, and every subset of a finite set is finite (The pigeonhole principle on $N$ , Statement; A subset of a finite set is finite, with $∣ B ∣ \leq ∣ A ∣$ , and equality holds if and only if $B = A$ , Statement; The cardinality $∣ A ∣$ of a finite set, Definition).

Proof

technique · explicit family of unit directions and comparison of the resulting great circles

1.1F1F2F3F4

Fix a unit $v \in T_{p} S_{R}^{n}$ . By [F1], $c_{p} (v) = π R$ is finite and $Cut (p) = {- p}$ . By [F2] the cut point of $p$ along $γ_{v}$ is $γ_{v} (π R)$ , so $γ_{v} (π R) \in Cut (p) = {- p}$ , and $d_{g} (p, γ_{v} (π R)) = π R$ . By [F3] the curve $γ_{v}$ is the maximal geodesic with $γ_{v} (0) = p$ and ${\dot{γ}}_{v} (0) = v$ , and by [F4] it has unit speed, so $L_{g} (γ_{v} ∣_{[0, π R]}) = π R = d_{g} (p, γ_{v} (π R))$ and the segment is a minimizing geodesic from $p$ to $- p$ .

1.2F6F7

By [F6] choose a chart at $p$ ; its coordinate derivations $\partial_{1} ∣_{p}, \dots, \partial_{n} ∣_{p}$ form a basis of $T_{p} S_{R}^{n}$ . The first two of these form a linearly independent list, so Gram--Schmidt [F7] supplies orthonormal $e_{1}, e_{2} \in T_{p} S_{R}^{n}$ with $g_{p} (e_{i}, e_{j}) = δ_{i j}$ .

2.1F5step 1.1

By [F5], for every unit $v$ the point $- p = γ_{v} (π R)$ is conjugate to $p$ along $γ_{v}$ with multiplicity $n - 1$ . Thus every minimizing meridian of step 1.1 ends at a conjugate point, and the conjugacy hypothesis of the refuted claim is met by each of them.

2.2F8F9step 1.2

For $m \in N$ define $u_{m} = \frac{e_{1} + m e_{2}}{\sqrt{1 + m^{2}}} \in T_{p} S_{R}^{n},$ a linear combination of tangent vectors. By [F8] the metric is bilinear and symmetric, so orthonormality in [F7] gives $g_{p} (e_{1} + m e_{2}, e_{1} + m e_{2}) = g_{p} (e_{1}, e_{1}) + 2 m g_{p} (e_{1}, e_{2}) + m^{2} g_{p} (e_{2}, e_{2}) = 1 + m^{2},$ and then, using $(\sqrt{1 + m^{2}})^{2} = 1 + m^{2}$ and $1 + m^{2} > 0$ from [F9], $g_{p} (u_{m}, u_{m}) = \frac{1 + m^{2}}{(\sqrt{1 + m^{2}})^{2}} = 1.$ So every $u_{m}$ is a unit tangent vector at $p$ .

3.1F9step 1.2step 2.2

Suppose $u_{m} = u_{k}$ with $m, k \in N$ . Since $e_{1}, e_{2}$ are orthonormal they are linearly independent, so the coefficients of a vector in their span are unique; comparing the coefficients of $e_{1}$ in $u_{m} = \frac{1}{\sqrt{1 + m^{2}}} e_{1} + \frac{m}{\sqrt{1 + m^{2}}} e_{2}$ and in the same expression with $k$ gives $\frac{1}{\sqrt{1 + m^{2}}} = \frac{1}{\sqrt{1 + k^{2}}},$ hence $\sqrt{1 + m^{2}} = \sqrt{1 + k^{2}}$ and, squaring and using [F9], $1 + m^{2} = 1 + k^{2}$ , that is $m^{2} = k^{2}$ . Since $m, k \geq 0$ in $R$ by [F9], the equivalence $a \leq b ⟺ a^{2} \leq b^{2}$ on nonnegative reals gives $m \leq k$ and $k \leq m$ , so $m = k$ . Therefore $m \mapsto u_{m}$ is injective.

4.1F3step 2.2step 3.1

For each $m \in N$ put $γ_{m} : = γ_{u_{m}}$ on $[0, π R]$ ; by [F3], $γ_{m} (0) = p$ and ${\dot{γ}}_{m} (0) = u_{m}$ . If $γ_{m} = γ_{k}$ as maps, then their derivatives at $t = 0$ coincide, so $u_{m} = u_{k}$ ; by step 3.1 this forces $m = k$ . Hence $m \mapsto γ_{m}$ is injective, and the meridians $γ_{m}$ are pairwise distinct.

5.1F10step 1.1step 2.1step 4.1

By steps 1.1 and 2.1 every $γ_{m}$ is a minimizing geodesic from $p$ to $- p$ along which $- p$ is conjugate to $p$ with multiplicity $n - 1$ , and by step 4.1 the members of the family are pairwise distinct. The set $G$ of minimizing geodesic segments from $p$ to $- p$ is infinite: if $G$ were finite, then its subset $G^{'} = {γ_{m} : m \in N}$ would be finite by [F10], so $G^{'} \approx ∣ G^{'} ∣$ with $∣ G^{'} ∣ \in N$ by [F10]; but $m \mapsto γ_{m}$ is a bijection $N \to G^{'}$ by step 4.1, hence $N \approx ∣ G^{'} ∣$ , contradicting the statement of [F10] that $N$ is not equinumerous with any natural number. Thus the conjugate point $- p$ is joined to $p$ by infinitely many distinct minimizing geodesics, and the claim in the Statement refuted is false: no meridian $γ_{m}$ is the unique minimizing geodesic from $p$ to the conjugate point $q = - p$ .

6.1A1F1F5F9F10step 1.1step 2.2step 4.1step 5.1

Boundary and choice audit. The dimension hypothesis $n \geq 2$ is used exactly at steps 1.2 and 2.2 to obtain two orthonormal tangent directions and a one-parameter family of unit directions; dimensions $n = 0$ (where $T_{p} S_{R}^{0}$ is the zero space, so no unit direction exists) and $n = 1$ (where the antipode has only the two semicircles) are outside the quantified claim, and nothing is asserted for them. The zero vector is never used: all $u_{m}$ , including $u_{0} = e_{1}$ , are unit vectors by step 2.2, and the family is still injective at $m = 0$ by step 3.1. The witness is nonempty because $p$ is supplied, and $p \neq - p$ because $∣ p ∣ = R > 0$ ; the interval $[0, π R]$ is nondegenerate because $R > 0$ , and the meridians are nonconstant geodesics of positive length. Both endpoints of $[0, π R]$ are included and the derivative at $0$ is one-sided for the injectivity argument of step 4.1. Exactly the inherited ${A C}_{ω}$ is assumed: it is spent only through the two sphere examples quoted in [F1] and [F5], while the orthonormal pair of step 1.2 comes from one finite basis list and the family of step 2.2 is given by an explicit formula, so no choice function on a family of directions is invoked. The refuted claim is a one-way uniqueness implication, so no converse case arises.

$□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed pp.189--190, discusses conjugate points and states (printed p.190) that no geodesic wrapping more than halfway around the flat cylinder is minimizing, and develops the cut locus defined by the last minimizing instant; the round-sphere antipodal geometry used here is the standard companion example. The explicit infinite family of minimizing meridians, its distinctness, and the refutation of uniqueness at the conjugate point are derived above from the pair's own sphere examples Cut locus of a point on a round sphere and Conjugate antipodes on the round sphere, not quoted from the source.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Distance hessian in euclidean space

Example

Let $n \geq 2$ and let $g_{E}$ be the standard Euclidean metric on $R^{n}$ . For $p \in R^{n}$ let $r_{p} (x) : = d_{g} (p, x)$ be the Riemannian distance from $p$ and let $\nabla^{2} r_{p} = \nabla (d r_{p})$ be its Levi-Civita covariant Hessian. Then, off $p$ , $\nabla^{2} r_{p} = \frac{1}{r_{p}} (g_{E} - d r_{p} \otimes d r_{p}),$ that is, for every $q \neq p$ and all $X, Y \in T_{q} R^{n}$ , $(\nabla^{2} r_{p})_{q} (X, Y) = \frac{1}{r_{p} (q)} (g_{E, q} (X, Y) - d r_{p} (X) d r_{p} (Y)),$ where $(d r_{p} \otimes d r_{p}) (X, Y) : = d r_{p} (X) d r_{p} (Y)$ . The proof shows in addition that $c_{p} (u) = + \infty$ for every unit direction $u \in S_{p} R^{n}$ , so every straight ray from $p$ minimizes for all time.

Facts & Assumptions

Given: An integer $n \geq 2$ , the Euclidean space $R^{n}$ with its standard metric $g_{E}$ , a point $p \in R^{n}$ , and a point $q \in R^{n}$ with $q \neq p$ .

[A1]

The Axiom of Countable Choice ${A C}_{ω}$ is the standing assumption (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

For every $n$ , $R^{n}$ is a smooth $n$ -manifold with the identity as global chart, without boundary; the standard Euclidean metric $g_{E} = \sum_{i = 1}^{n} d x^{i} \otimes d x^{i}$ has Cartesian metric matrix $(δ_{i j})$ , which is smooth, symmetric and positive definite, so $(R^{n}, g_{E})$ is a Riemannian manifold (Euclidean spaces and Euclidean open subsets as smooth manifolds, Euclidean space has zero curvature, Coordinate criterion for a riemannian metric, Riemannian metric and riemannian manifold).

[F2]

In the global Cartesian chart the metric matrix of $g_{E}$ is $(δ_{i j})$ , so the coordinate derivations $\partial_{1} ∣_{a}, \dots, \partial_{n} ∣_{a}$ form a basis of $T_{a} R^{n}$ and $g_{E, a} (v, v) = \sum_{i} (v^{i})^{2}$ for $v = \sum_{i} v^{i} \partial_{i} ∣_{a}$ ; hence $∣ v ∣_{g_{E}} = ∥ (v^{1}, \dots, v^{n}) ∥_{2}$ is the Euclidean norm of the coefficient vector. The curve $c (t) = a + t w$ has velocity $\dot{c} (t) = \sum_{i} w^{i} \partial_{i} ∣_{c (t)}$ : the velocity derivation acts on a smooth germ $f$ by $(f \circ c)^{'} (t) = D_{w} f (c (t)) = ⟨ \nabla f (c (t)), w ⟩ = \sum_{i} w^{i} \partial_{i} f (c (t))$ . Consequently the straight line $t \mapsto a + t w$ has the constant $g_{E}$ -speed $∣ \dot{c} ∣_{g_{E}} = ∥ w ∥_{2}$ (The euclidean metric and its musical maps, Coordinate criterion for a riemannian metric, Coordinate derivations form a basis of the tangent space, Pointwise norm and angle from a riemannian metric, The velocity derivation of a smooth curve, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , For a differentiable scalar field, $D_{v} f (a) = ⟨ \nabla f (a), v ⟩$ and the unit direction of steepest ascent is the normalized gradient, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F3]

For the Euclidean inner product, $⟨ z, z ⟩ \geq 0$ for every $z \in R^{n}$ , with equality exactly when $z = 0$ ; hence the norm $∥ z ∥_{2} = \sqrt{⟨ z, z ⟩}$ is nonnegative and vanishes exactly at $z = 0$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

On $R^{n}$ with $g_{E}$ , a geodesic on an interval $I$ is exactly a curve of the form $γ (t) = x + t v$ with constant $x, v \in R^{n}$ , and every such curve is a geodesic with velocity $v$ (Straight lines as Euclidean geodesics).

[F5]

Under [A1], for every $(x, v)$ there is a unique maximal geodesic $γ_{x, v}$ with $γ_{x, v} (0) = x$ and $γ_{x, v}^{'} (0) = v$ , defined on an open interval containing $0$ ; the exponential map is $\exp_{x} (v) = γ_{x, v} (1)$ (Existence uniqueness and smooth dependence of geodesics, Domain and exponential map of a connection).

[F6]

A boundaryless Riemannian manifold is geodesically complete when every maximal geodesic has domain $R$ ; the zero initial vector is included (Geodesically complete Riemannian manifold).

[F7]

On a $C^{1}$ piece the Riemannian speed is $∣ \dot{γ} ∣_{g}$ and the length of a piecewise $C^{1}$ curve is $L_{g} (γ) = \sum_{j} \int ∣ \dot{γ} ∣_{g} d t$ , the empty sum and a constant curve giving zero; on a connected Riemannian manifold the distance is $d_{g} (p, x) = \inf {L_{g} (γ) : γ$ piecewise $C^{1}$ from $p$ to $x}$ , a finite nonnegative infimum, and no minimizing curve is part of the definition (Riemannian speed and length, Piecewise c one curve on a manifold, Riemannian distance on a connected manifold).

[F8]

For a piecewise $C^{1}$ path $c : [a, b] \to R^{n}$ — under the identity global chart these are exactly the piecewise $C^{1}$ curves into the manifold $R^{n}$ , with the same speed — the polygonal arc length is $L (c) = \sum_{j} \int ∥ c^{'} (t) ∥_{2} d t$ ; moreover $∥ c (v) - c (u) ∥_{2} \leq L (c ∣_{[u, v]})$ for $a \leq u \leq v \leq b$ (A continuous piecewise- $C^{1}$ path is rectifiable and its length is the sum of the speed integrals over its pieces, Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability, Every endpoint chord is no longer than the arc: $∥ γ (b) - γ (a) ∥_{2} \leq L (γ)$ ).

[F9]

If $ℓ$ is a real lower bound of a nonempty set $S \subseteq R$ of lengths admitting a finite infimum, then $ℓ \leq \inf S$ (Greatest lower bound (infimum), Lower bound, bounded below, bounded set).

[F10]

For $n \geq 1$ , $R^{n}$ is polygonally connected and connected ( $R^{n}$ is polygonally connected, connected, locally path-connected and locally connected).

[F11]

Every constant function with value $c$ on $[a, b]$ is Darboux integrable with $\int_{a}^{b} c d t = c (b - a)$ ; in particular $\int_{0}^{s} 1 d t = s$ for $s > 0$ (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ).

[F12]

Under [A1], for a nonempty connected boundaryless Riemannian manifold, metric completeness of $(M, d_{g})$ , geodesic completeness, $E_{p} = T_{p} M$ for every $p$ , the same for one $p_{0}$ , and compactness of every closed bounded subset are equivalent; when they hold every two points are joined by a minimizing geodesic (Hopf–Rinow theorem).

[F13]

Under [A1], for a complete connected boundaryless Riemannian manifold $(M, g)$ , $p \in M$ and a unit $v \in T_{p} M$ , the cut time is $c_{p} (v) = \sup {t > 0 : d_{g} (p, \exp_{p} (t v)) = t} \in (0, + \infty]$ , and the value $+ \infty$ is allowed when every positive radial segment minimizes (Cut time in a unit tangent direction).

[F14]

A smooth field $J$ along $γ (t) = x + t v$ is a Jacobi field if and only if $J (t) = A + t B$ with constant $A, B \in R^{n}$ , in which case $D_{t} J = B$ (Jacobi fields in euclidean space).

[F15]

Under [A1], let $(M, g)$ be complete, connected and boundaryless, let $p \in M$ , let $v \in S_{p} M$ be unit, let $0 < t < c_{p} (v)$ , put $q = \exp_{p} (t v)$ and $T = \dot{γ} (t)$ . Then: (i) for every $X \in T_{q} M$ with $g_{q} (X, T) = 0$ there is exactly one Jacobi field $J_{X}$ along $γ$ with $J_{X} (0) = 0$ and $J_{X} (t) = X$ ; (ii) $(\nabla^{2} r_{p})_{q} (X, Y) = g_{q} (D_{t} J_{X} (t), Y)$ for such $X$ and every $Y$ ; (iii) the endomorphism $S = \nabla grad r_{p}$ satisfies $g_{q} (S Z, W) = (\nabla^{2} r_{p})_{q} (Z, W)$ and $S (T) = 0$ , while $S (X) = D_{t} J_{X} (t)$ for every $X ⊥ T$ (Hessian of distance in terms of radial jacobi fields, Gradient hessian and divergence connection formulas).

[F16]

Under [A1], for complete connected boundaryless $(M, g)$ , $p \in M$ , unit $v \in S_{p} M$ and $0 < t < c_{p} (v)$ , with $q = \exp_{p} (t v)$ : $grad r_{p} (q) = \dot{γ} (t) = γ^{'} (t)$ and $∣ grad r_{p} (q) ∣_{g} = 1$ (Gradient of the distance is the outward unit radial field off the base point and the cut locus).

[F17]

Here the Riemannian gradient means the unique vector field characterized by $g_{x} ((grad f)_{x}, Z) = d f_{x} (Z)$ for every smooth real $f$ , every $x$ and every $Z \in T_{x} M$ ; this is the defining identity used in the calculations below.

Verification

technique · direct: compute the Euclidean Riemannian distance and cut time, then read the Hessian off the radial Jacobi fields of the straight geodesic

1.1F1F2F3F4

The radial direction. Put $ρ : = ∥ q - p ∥_{2}$ and $u : = (q - p) / ρ$ . By [F3] $ρ > 0$ , and $q = p + ρ u$ with $∥ u ∥_{2} = 1$ . Under the canonical identification of [F2], $u$ is a unit tangent vector at $p$ , $∣ u ∣_{g_{E}} = 1$ , and $γ (t) : = p + t u$ is, by [F4], the geodesic with $γ (0) = p$ and $\dot{γ} (0) = u$ ; its velocity is the constant vector $u$ , so $∣ \dot{γ} ∣_{g_{E}} = 1$ and $T : = \dot{γ} (ρ) = u$ has $g_{E, q} (T, T) = 1$ .

1.2F2F7F8F9F11

The Euclidean Riemannian distance. Let $x \in R^{n}$ . If $x = p$ both sides of $d_{g} (p, x) = ∥ x - p ∥_{2}$ are $0$ , because the constant curve has length $0$ [F7]. Let $x \neq p$ , put $s : = ∥ x - p ∥_{2} > 0$ and $w : = (x - p) / s$ , so $∥ w ∥_{2} = 1$ and $x = p + s w$ . The segment $c (t) = p + t w$ , $t \in [0, s]$ , is a piecewise $C^{1}$ curve from $p$ to $x$ of constant speed $∣ \dot{c} ∣_{g_{E}} = ∥ w ∥_{2} = 1$ by [F2], so [F7] and [F11] give $L_{g} (c) = \int_{0}^{s} 1 d t = s$ ; hence $d_{g} (p, x) \leq s$ . Conversely, for an arbitrary piecewise $C^{1}$ curve $c$ from $p$ to $x$ , [F2] and [F7] identify its Riemannian length with $\sum_{j} \int ∥ c^{'} (t) ∥_{2} d t$ , which is the polygonal arc length $L (c)$ of [F8], and the chord bound of [F8] gives $L_{g} (c) = L (c) \geq ∥ x - p ∥_{2} = s$ . Thus $s$ is a real lower bound of the nonempty set of curve lengths whose infimum is $d_{g} (p, x)$ [F7], so [F9] gives $s \leq d_{g} (p, x)$ . Therefore $d_{g} (p, x) = ∥ x - p ∥_{2} (x \in R^{n}) .$

1.3F4F5F6F10F12

Geodesic completeness and the exponential map. By [F1] and [F10] the manifold $(R^{n}, g_{E})$ is a boundaryless connected Riemannian manifold. Let $γ_{x, v}$ be the maximal geodesic with $γ_{x, v} (0) = x$ and $γ_{x, v}^{'} (0) = v$ on its open domain $I$ [F5]. On $I$ , [F4] writes $γ_{x, v} (t) = a + t b$ with constant $a, b$ , so $a = γ_{x, v} (0) = x$ and $b = γ_{x, v}^{'} (0) = v$ , that is $γ_{x, v} (t) = x + t v$ . The curve $ℓ (t) = x + t v$ , $t \in R$ , is a geodesic by [F4] with the same initial data; since its domain cannot be enlarged to a longer interval, it is a maximal geodesic with these data, and the uniqueness in [F5] forces $γ_{x, v} = ℓ$ and $I = R$ . Hence every maximal geodesic has domain $R$ and $(R^{n}, g_{E})$ is geodesically complete [F6]; by the equivalence of [F12] it satisfies the completeness hypothesis used by the cut-time and distance-Hessian results, and in particular $\exp_{x} (w) = γ_{x, w} (1) = x + w (w \in T_{x} R^{n} ≅ R^{n}) .$

2.1F12F13step 1.1step 1.2step 1.3

The cut time in every direction is infinite. Fix the unit direction $u$ of 1.1. For every $t > 0$ , 1.3 gives $\exp_{p} (t u) = p + t u$ , and 1.2 gives $d_{g} (p, \exp_{p} (t u)) = ∥ t u ∥_{2} = t$ (using $∥ u ∥_{2} = 1$ from 1.1). Hence the set ${t > 0 : d_{g} (p, \exp_{p} (t u)) = t}$ is all of $(0, \infty)$ ; since $(R^{n}, g_{E})$ is complete, connected and boundaryless by 1.3, the cut-time definition [F13] assigns $c_{p} (u) = + \infty$ , the value allowed there when every positive radial segment minimizes. In particular $0 < ρ < c_{p} (u)$ with $ρ > 0$ from 1.1.

3.1F4F14F15step 1.1step 2.1

The radial Jacobi fields. Let $X \in T_{q} R^{n}$ satisfy $g_{E, q} (X, T) = 0$ , with $T = u$ the unit vector of 1.1. By 1.1 and 2.1 the hypotheses of [F15] hold with $v = u$ and $t = ρ$ , so there is exactly one Jacobi field $J_{X}$ along $γ$ with $J_{X} (0) = 0$ and $J_{X} (ρ) = X$ . The field $J (s) : = (s / ρ) X$ — the vector $X$ being extended as a constant field in the Cartesian trivialization — has the affine form $A + s B$ with $A = 0$ and $B = X / ρ$ , so it is a Jacobi field by [F14] and satisfies $J (0) = 0$ , $J (ρ) = X$ ; by the uniqueness in F15, $J_{X} = J$ and $D_{s} J_{X} (ρ) = X / ρ .$ Consequently the endomorphism $S = \nabla grad r_{p}$ of F15 satisfies $S (T) = 0$ and $S (X) = X / ρ$ for every $X ⊥ T$ .

3.2F16F17step 2.1

The gradient of the distance. By [F16], applied with the unit direction $u$ and $0 < ρ < c_{p} (u)$ from 2.1, $grad r_{p} (q) = \dot{γ} (ρ) = T$ and $∣ grad r_{p} (q) ∣_{g_{E}} = 1$ . The gradient characterization [F17] therefore gives $d r_{p} (Z) = g_{E, q} (grad r_{p} (q), Z) = g_{E, q} (T, Z) (Z \in T_{q} R^{n}),$ and in particular $d r_{p} (T) = g_{E, q} (T, T) = 1$ .

4.1F15step 1.2step 3.1step 3.2

The Hessian identity. Let $Z \in T_{q} R^{n}$ and write $Z = Z^{⊥} + g_{E, q} (Z, T) T$ with $Z^{⊥} : = Z - g_{E, q} (Z, T) T$ orthogonal to $T$ ; this is the orthogonal decomposition because $g_{E, q} (T, T) = 1$ . By linearity of $S$ and step 3.1, $S (Z) = S (Z^{⊥}) = Z^{⊥} / ρ$ . Since $S$ represents the Hessian by F15, for every $W \in T_{q} R^{n}$ one has $(\nabla^{2} r_{p})_{q} (Z, W) = g_{E, q} (S (Z), W) = \frac{1}{ρ} g_{E, q} (Z^{⊥}, W) = \frac{1}{ρ} (g_{E, q} (Z, W) - g_{E, q} (Z, T) g_{E, q} (T, W)) = \frac{1}{ρ} (g_{E, q} (Z, W) - d r_{p} (Z) d r_{p} (W)),$ the third equality by bilinearity and the definition of $Z^{⊥}$ , the fourth by step 3.2. Since $r_{p} (q) = d_{g} (p, q) = ∥ q - p ∥_{2} = ρ$ by 1.2, this is the asserted identity at $q$ ; the point $q \neq p$ was arbitrary, so the identity holds at every point of $R^{n} ∖ {p}$ .

5.1A1F5F10F12F13F15F16step 1.1step 1.2step 1.3step 2.1step 3.1step 3.2step 4.1givencases∎

Audit. The hypothesis $n \geq 2$ of the statement is respected; the argument uses only $n \geq 1$ (for the connectedness supplied by [F10]) and no case outside the stated hypothesis is claimed. Since $q \neq p$ , [F3] gives $ρ > 0$ , so the divisions by $ρ$ and by $r_{p} (q) = ρ$ are legitimate; the point $p$ itself is excluded and no differentiability or Hessian value at $p$ is asserted. The zero vector is harmless: $Z = 0$ gives $S (0) = 0$ and both sides of the identity vanish, while the radial vector $T$ is unit. There is no endpoint in the parameter range: $0 < ρ < c_{p} (u) = + \infty$ by 2.1, so no cut point occurs along this ray and the formula is interior. All constructions are explicit: $u = (q - p) / ρ$ and $γ (t) = p + t u$ involve no selection, and [A1] is inherited exactly through the declared suppliers that carry it, namely the geodesic existence, uniqueness and smooth-dependence theorem [F5], the cut-time definition [F13], Hopf--Rinow [F12], and the Hessian and gradient results [F15, F16]. No biconditional is asserted: the conclusion is a tensor identity off $p$ , the distance computation of 1.2 is proved by the two inequalities, and the Jacobi classification [F14] is used in the direction from the affine form to the Jacobi equation.

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed pp.173-190, develops Jacobi fields, the identification of $d (\exp_{p})_{t v}$ with the endpoint of a Jacobi field vanishing at $p$ , and the Hessian of the distance function outside the cut locus; Chapter 5, printed p.81, records that the Euclidean geodesics are the straight lines. Datar, Lectures on Riemannian Geometry, Section 23.2, printed pp.167-170, states that $grad r_{p}$ is the unit radial field off the cut locus, and Section 23.3, printed pp.171-172, treats the regularity of the distance. Neither source writes out the Euclidean specialization $(\nabla^{2} r_{p})_{q} = (1 / r_{p} (q)) (g - d r_{p} \otimes d r_{p})$ ; the computation above derives it from the library's Euclidean-geometry, length-and-distance and distance-Hessian suppliers, and no source text is quoted.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Index form in constant curvature

Example

Assume exactly ${A C}_{ω}$ through the declared dependencies. Let $(M, g)$ be a finite-dimensional Riemannian manifold without boundary of constant sectional curvature $K \in R$ , let $a < b$ , and let $γ : [a, b] \to M$ be a unit-speed affinely parametrized geodesic. Write $T = \dot{γ}$ and, for a field $V$ along $γ$ , $V_{⊥} : = V - g (V, T) T .$

(a) For all continuous piecewise $C^{1}$ fields $V, W$ along $γ$ , $I_{γ} (V, W) = \int_{a}^{b} (g (D_{t} V, D_{t} W) - K g (V_{⊥}, W_{⊥})) d t,$ and in particular $I_{γ} (V, V) = \int_{a}^{b} (g (D_{t} V, D_{t} V) - K ∣ V_{⊥} ∣^{2}) d t$ . On the fixed-endpoint subspace this is the formula $I_{γ} (V, V) = \int_{a}^{b} (∣ D_{t} V ∣^{2} - K ∣ V_{⊥} ∣^{2}) d t$ .

(b) Suppose $K > 0$ , put $L : = π / \sqrt{K}$ , let $γ : [0, L] \to M$ be a unit-speed affinely parametrized geodesic, and let $E$ be a parallel normal field along $γ$ with $g (E, E) = 1$ . Such a field $E$ is data of the example and no existence claim is made here. Put $s_{K} (t) = \frac{\sin (\sqrt{K} t)}{\sqrt{K}}, J (t) : = s_{K} (t) E (t) .$ Then $J$ is a nonzero Jacobi field with $J (0) = J (L) = 0$ , and $I_{γ} (J, W) = 0$ for every $W \in X_{0} (γ)$ ; in particular $I_{γ} (J, J) = 0$ , so the fixed-endpoint index form on $[0, L]$ is degenerate.

(c) With the same data and any $L > π / \sqrt{K}$ , the field $W (t) = \sin (π t / L) E (t)$ is smooth, lies in $X_{0} (γ)$ , and $I_{γ} (W, W) = (\frac{π^{2}}{L^{2}} - K) \int_{0}^{L} \sin^{2} (\frac{π t}{L}) d t < 0.$ So on a unit-speed geodesic segment of length $L > π / \sqrt{K}$ the fixed-endpoint index form is negative on a field and is not positive semidefinite.

Facts & Assumptions

Given: The constant-curvature manifold $(M, g)$ , the unit-speed affine geodesic $γ$ on its nondegenerate interval, a field $V$ as in (a), the positive curvature $K > 0$ and the length $L = π / \sqrt{K}$ in (b), and, in (b) and (c), a parallel normal unit field $E$ along $γ$ .

[A1]

Countable choice is the assumption ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ). It is inherited here through the index-form and constant-curvature interfaces (Index form of a geodesic segment, Curvature tensor of constant sectional curvature, Jacobi fields in constant sectional curvature). The pointwise algebra, the trigonometric computations and the two test fields use no further choice, and no full Axiom of Choice is assumed.

[F1]

Constant sectional curvature $K$ means that every tangent two-plane has sectional curvature $K$ ; in dimensions zero and one the predicate is vacuous for every $K$ (Constant sectional curvature and space form).

[F2]

Under this hypothesis the curvature operator is $R (X, Y) Z = K (g (Y, Z) X - g (X, Z) Y)$ (Curvature tensor of constant sectional curvature).

[F3]

The index form of a geodesic segment on the space of continuous piecewise $C^{1}$ fields is the finite sum of the integrals of $g (D_{t} V, D_{t} W) - g (R (V, \dot{γ}) \dot{γ}, W)$ , and $X_{0} (γ)$ is its fixed-endpoint subspace (Index form of a geodesic segment).

[F4]

The metric $g$ is symmetric, bilinear and positive definite, so $∣ U ∣^{2} = g (U, U) \geq 0$ with equality exactly for $U = 0$ (Riemannian metric and riemannian manifold).

[F5]

$D_{t}$ and $D_{t}^{2}$ are the covariant derivatives along $γ$ , with one-sided values at an included endpoint (Covariant derivative along a curve).

[F6]

A parallel field satisfies $D_{t} E = 0$ , and $E$ is normal when $g (E, T) = 0$ (Parallel section along a curve).

[F7]

For $K > 0$ the normal Jacobi fields $J$ along a unit-speed geodesic with $J (0) = 0$ are exactly the fields $s_{K} (t) E (t)$ with $E$ a parallel normal field; the exponential factor is $s_{K} (t) = \sin (\sqrt{K} t) / \sqrt{K}$ (Jacobi fields in constant sectional curvature).

[F8]

A Jacobi field satisfies $D_{t}^{2} J + R (J, T) T = 0$ on the interval (Jacobi field).

[F9]

If $V$ is $C^{2}$ and $W$ is $C^{1}$ on the pieces of a finite subdivision, then $I_{γ} (V, W) = [g (D_{t} V, W)]_{a}^{b}$ minus the derivative-jump terms minus $\int_{a}^{b} g (D_{t}^{2} V + R (V, T) T, W) d t$ , the jumps being absent when both fields are smooth (Integration by parts for the index form).

[F10]

$\sin π = 0$ , and $\sin x > 0$ for $0 < x < π$ (Pi is the first positive zero of sine).

[F11]

$(\sin x)^{'} = \cos x$ , $(\cos x)^{'} = - \sin x$ and $\sin 0 = 0$ (The derivatives of sine and cosine are cosine and minus sine); the chain rule gives the derivatives of $t \mapsto \sin (α t)$ for constant $α$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F12]

For an integrable $h$ and real $λ$ , $\int_{a}^{b} λ h = λ \int_{a}^{b} h$ (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ).

[F13]

A nonnegative integrable function has nonnegative integral (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[F14]

A continuous nonnegative function on $[a, b]$ with integral $0$ vanishes identically (A continuous $f \geq 0$ on $[a, b]$ with $\int_{a}^{b} f = 0$ is identically $0$ ).

Verification

Proof technique: insert the constant-curvature operator into the index form, then evaluate the sine solution and a sine test field explicitly.

1.1

For every field $V$ along $γ$ and every $t$ one has $g (V_{⊥}, T) = 0$ and $R (V, T) T = K V_{⊥}$ . [F2, F4, given] By definition of $V_{⊥}$ and bilinearity of $g$ , $g (V_{⊥}, T) = g (V, T) - g (V, T) g (T, T) = 0$ because $γ$ has unit speed, $g (T, T) = 1$ . Then [F2] gives $R (V, T) T = K (g (T, T) V - g (V, T) T) = K (V - g (V, T) T) = K V_{⊥}$ .

1.2

In the setting of (b), the field $J (t) = s_{K} (t) E (t)$ is a nonzero Jacobi field with $J (0) = J (L) = 0$ . [F6, F7, F10, F11] The field $E$ is parallel and normal by [F6], so [F7] identifies $s_{K} E$ as a normal Jacobi field with $J (0) = 0$ . Since $s_{K} (0) = \sin 0 / \sqrt{K} = 0$ and $s_{K} (L) = \sin (π) / \sqrt{K} = 0$ by [F11] and [F10], both endpoint values vanish; and $s_{K} (t) > 0$ for $0 < t < L$ by [F10] while $∣ E (t) ∣ = 1$ , so $J$ is not the zero field.

1.3

Let $φ (t) = \sin (π t / L)$ for $0 \leq t \leq L$ . Then $φ$ is smooth with $φ (0) = φ (L) = 0$ and $φ^{''} (t) = - (π / L)^{2} φ (t)$ . [F10, F11] The chain rule [F11] gives $φ^{'} (t) = (π / L) \cos (π t / L)$ and $φ^{''} (t) = - (π / L)^{2} \sin (π t / L)$ . The endpoints use $\sin 0 = 0$ and $\sin π = 0$ .

2.1

For all continuous piecewise $C^{1}$ fields $V, W$ along $γ$ , the index form is $I_{γ} (V, W) = \int_{a}^{b} (g (D_{t} V, D_{t} W) - K g (V_{⊥}, W_{⊥})) d t$ , and on the diagonal $I_{γ} (V, V) = \int_{a}^{b} (g (D_{t} V, D_{t} V) - K ∣ V_{⊥} ∣^{2}) d t$ . [F3, F4, step 1.1] By step 1.1, $R (V, T) T = K V_{⊥}$ at every $t$ ; moreover $g (V_{⊥}, W) = g (V_{⊥}, W_{⊥})$ because $W - W_{⊥} = g (W, T) T$ is parallel to $T$ and $g (V_{⊥}, T) = 0$ . Substituting into the defining sum of [F3] and using the diagonal identification $g (V_{⊥}, V_{⊥}) = ∣ V_{⊥} ∣^{2}$ of [F4] gives both displayed identities, in particular on $X_{0} (γ)$ .

2.2

$I_{γ} (J, W) = 0$ for every $W \in X_{0} (γ)$ . [F5, F8, F9, step 1.2] Both $J$ and $W$ are continuous on $[0, L]$ , with $J$ smooth and $W$ piecewise $C^{1}$ , so [F9] applies with no derivative jumps: the boundary term $[g (D_{t} J, W)]_{0}^{L}$ vanishes because $W (0) = W (L) = 0$ , and the integral term vanishes pointwise because $D_{t}^{2} J + R (J, T) T = 0$ by [F8] and step 1.2. Since $J$ is a nonzero element of the subspace $X_{0} (γ)$ , the fixed-endpoint index form on $[0, L]$ is degenerate.

2.3

In the setting of (c), $W (t) = φ (t) E (t)$ lies in $X_{0} (γ)$ and $I_{γ} (W, W) = ((π / L)^{2} - K) \int_{0}^{L} φ^{2} d t$ . [F2, F5, F6, F9, F12, step 1.3] The field $W$ is smooth because $φ$ and $E$ are, and $W (0) = W (L) = 0$ by step 1.3, so $W \in X_{0} (γ)$ . Since $D_{t} E = 0$ by [F6], one has $D_{t} W = φ^{'} E$ and $D_{t}^{2} W = φ^{''} E$ ; and $R (W, T) T = K (g (T, T) W - g (W, T) T) = K φ E$ by [F2], because $g (E, T) = 0$ and $g (T, T) = 1$ . So $D_{t}^{2} W + R (W, T) T = (φ^{''} + K φ) E$ , and the pairing with $W$ is $(φ^{''} + K φ) φ$ . Applying [F9] with $V = W$ and no jumps, the boundary term is $[g (D_{t} W, W)]_{0}^{L} = [φ^{'} φ ∣ E ∣^{2}]_{0}^{L} = 0$ because $φ (0) = φ (L) = 0$ , hence $I_{γ} (W, W) = - \int_{0}^{L} (φ^{''} + K φ) φ d t$ . Step 1.3 gives $φ^{''} + K φ = (K - (π / L)^{2}) φ$ , so the last integral equals $(K - (π / L)^{2}) \int_{0}^{L} φ^{2} d t$ by [F12], and its negative is the displayed value.

2.4

In the setting of (c), $\int_{0}^{L} φ^{2} d t > 0$ . [F10, F13, F14, step 1.3] The function $φ^{2}$ is continuous and nonnegative on $[0, L]$ . At the midpoint $t = L / 2$ one has $φ (L / 2) = \sin (π / 2) > 0$ by [F10], so $φ^{2}$ is not identically zero; if its integral vanished, [F14] would force $φ \equiv 0$ , a contradiction. Therefore the integral is nonzero, and it is $\geq 0$ by [F13]; hence it is strictly positive.

3.1

If $L > π / \sqrt{K}$ then $I_{γ} (W, W) < 0$ . [step 2.3, step 2.4, algebra] By step 2.3, $I_{γ} (W, W) = ((π / L)^{2} - K) \int_{0}^{L} φ^{2} d t$ . The assumption $L > π / \sqrt{K}$ makes $(π / L)^{2} - K < 0$ , and step 2.4 makes the second factor positive, so the product is negative. Therefore the fixed-endpoint index form on a unit-speed segment of length $L > π / \sqrt{K}$ is not positive semidefinite.

4.1

Boundary, degeneracy and choice audit. [A1, F1, F3, F5, F9, step 1.1, step 2.1, step 2.2, step 3.1] The interval $[a, b]$ of (a) is nondegenerate and included endpoints carry the one-sided derivatives of [F5]. In (b) and (c) the length $L > 0$ is fixed and $0$ is an included endpoint; at the critical length $L = π / \sqrt{K}$ the case (c) is excluded and case (b) shows only degeneracy, not negativity. The formulas are vacuous in the empty manifold and in dimension zero, where no unit-speed geodesic exists; in dimension one every field is parallel to $T$ , so $V_{⊥} = 0$ , part (a) reduces to $I_{γ} (V, V) = \int ∣ D_{t} V ∣^{2} d t$ , and parts (b) and (c) are vacuous because no normal direction exists. For $K = 0$ part (a) gives the flat formula $I_{γ} (V, W) = \int g (D_{t} V, D_{t} W) d t$ , while (b) and (c) require $K > 0$ . The zero field has zero index form, and the nonzero field $J$ of step 2.2 is the explicit degeneracy witness. Assumption [A1] is inherited from the index-form and constant-curvature suppliers; the two test fields are given by explicit formulas and no selection is made. The example asserts identities and one-way implications and claims no equivalence. $□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, equation (10.15) and the surrounding index-form discussion, printed p.186 / PDF label P203, states the index form on proper normal fields and computes its curvature term in constant curvature; Datar, Lectures on Riemannian Geometry, §21.2 (printed pp.156–157) and §23.1–23.3 (printed pp.165–169), treats the same form and its positivity below the first conjugate point. The reduction to $K g (V_{⊥}, W_{⊥})$ , the sine zero mode and the negative sine test field are computed above rather than quoted.

Jacobi Fields, Conjugate Points, and the Cut Locus — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Jacobi fields in euclidean space

Example

Facts & Assumptions

Proof

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Jacobi fields in constant sectional curvature

Example

Facts & Assumptions

Verification

Source notes

Conjugate antipodes on the round sphere

Example

Facts & Assumptions

Verification

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No conjugate points in nonpositive constant curvature

Example

Facts & Assumptions

Verification

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Killing jacobi fields from rotations

Example

Facts & Assumptions

Verification

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Cut locus of a point on a round sphere

Example

Facts & Assumptions

Verification

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Cut locus of a point on a flat circle

Example

Facts & Assumptions

Proof

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Cut locus on a flat rectangular torus from the Dirichlet cell

Example

Facts & Assumptions

Proof

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A cut point that is not conjugate because two minimizers arrive

Statement refuted

Facts & Assumptions

Proof

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A conjugate point at which there are many geodesics

Statement refuted

Facts & Assumptions

Proof

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Distance hessian in euclidean space

Example

Facts & Assumptions

Verification

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Index form in constant curvature

Example

Facts & Assumptions

Verification

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Sources