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Model jacobi fields in positive zero and negative curvature

Example

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

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

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

Facts & Assumptions

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

[A1]

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

[F1]

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

[F2]

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

[F3]

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

[F4]

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

[F5]

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

Verification

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

1.1F1F3given

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

2.1F2F3F4step 1.1

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

3.1F3step 2.1

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

4.1F2F4step 3.1

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

5.1F3F5step 1.1step 3.1step 4.1∎

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

Source locator

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

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Sources

Ved Datar, Lectures on Riemannian Geometry (2025) (standard reference, not scraped)
J.-H. Eschenburg, Comparison Theorems in Riemannian Geometry (standard reference, not scraped)