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Higher sectional curvature makes jacobi fields spread faster

Statement

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ .

False claim. Suppose two unit-speed geodesics $γ_{1}, γ_{2}$ in $n$ -dimensional Riemannian manifolds have normal Jacobi fields $J_{1}, J_{2}$ with $J_{i} (0) = 0$ and equal positive initial-derivative norms, and suppose every radial sectional curvature of the first manifold is at least every radial sectional curvature of the second. Then higher curvature makes the field longer: $∣ J_{1} (t) ∣ \geq ∣ J_{2} (t) ∣$ at every time $t$ in the common interval of definition.

The claim is false. On a segment $[0, T]$ where $γ_{1} (0)$ has no conjugate point along $γ_{1}$ in $(0, T]$ , Rauch's first comparison (Rauch comparison theorem first form) gives $∣ J_{1} (t) ∣ \leq ∣ J_{2} (t) ∣$ for $0 \leq t \leq T$ . This reversed comparison is restricted to that segment; it is not asserted beyond the first conjugate instant of the more-curved geodesic.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1] and the false claim above; the refutation uses the explicit comparison pair below.

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), inherited through the constant-sectional-curvature and curvature-tensor interfaces below; the parallel initial-value supplier itself is choice-free.

[F1]

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

[F2]

Model geometry (Constant sectional curvature and space form, Curvature tensor of constant sectional curvature, Jacobi field): a manifold of constant sectional curvature $k$ has $R (X, Y) Z = k (g (Y, Z) X - g (X, Z) Y)$ ; the unit sphere has $k = 1$ and Euclidean space has $k = 0$ , so on a unit-speed geodesic $γ$ and for a parallel normal unit field $Φ_{t} E$ the fields $J_{s p h e r e} (t) = \sin t Φ_{t} E, J_{e u c l i d} (t) = t Φ_{t} E$ satisfy $D_{t}^{2} J + R (J, \dot{γ}) \dot{γ} = 0$ : indeed $J_{s p h e r e}^{''} = - J_{s p h e r e} = - R (J_{s p h e r e}, \dot{γ}) \dot{γ}$ on the unit sphere and $J_{e u c l i d}^{''} = 0 = R (J_{e u c l i d}, \dot{γ}) \dot{γ}$ in Euclidean space.

[F3]

Parallel transport (Existence and uniqueness of parallel sections, Levi civita parallel transport preserves lengths angles and volume): the parallel field with prescribed unit normal value exists, is unique and has constant norm $1$ .

[F4]

Rauch comparison, first form (Rauch comparison theorem first form): under its two hypotheses the inequality is $∣ J_{1} ∣ \leq ∣ J_{2} ∣$ when the first manifold is the more curved one.

[F5]

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

Refutation

Proof technique: direct: exhibit the sphere-versus-Euclidean pair, check the hypotheses of the false claim, and evaluate the asserted inequality at a single time where the reverse strict inequality holds.

1.1F1F2F3given

The witness pair. [F1, F2, F3, given] Take the unit sphere $S^{n}$ with its round metric of curvature $1$ and Euclidean space $R^{n}$ , both of dimension $n \geq 2$ . Let $γ_{1}$ be a unit-speed great-circle geodesic on $S^{n}$ and $γ_{2}$ a unit-speed straight line in $R^{n}$ ; let $E \in T_{γ_{i} (0)} M_{i}$ be a unit normal vector and $Φ_{t} E$ its parallel transport along $γ_{i}$ [F3]. Restrict the witness pair to $[0, 1]$ . Put $J_{1} (t) : = {sn}_{1} (t) Φ_{t} E = \sin t Φ_{t} E, J_{2} (t) : = t Φ_{t} E .$ By [F1] and [F2] both are normal Jacobi fields with $J_{i} (0) = 0, ∣ D_{t} J_{i} (0) ∣ = 1,$ so the initial data have the equal positive norm required by the false claim; and $∣ J_{1} (t) ∣ = \sin t, ∣ J_{2} (t) ∣ = t .$

2.1F1F2F4step 1.1given

The curvature hypothesis holds and the true inequality is reversed. [F1, F2, F4, step 1.1, given] Every radial sectional curvature of the unit sphere is $1$ , and every sectional curvature of Euclidean space is $0$ , by [F2]; hence every radial curvature of the first manifold is at least every radial curvature of the second. For this explicit pair, [F5] gives $\sin t \leq t$ on $[0, 1]$ , so directly $∣ J_{1} (t) ∣ \leq ∣ J_{2} (t) ∣ (0 \leq t \leq 1) .$ This calculation refutes the proposed direction without needing to apply Rauch or verify its no-conjugate hypothesis.

3.1F5step 1.1step 2.1given∎

The false claim fails at $t = 1$ . [F5, step 1.1, step 2.1, given] By [F5] with $t = 1 \leq 2$ , $\sin 1 = \cos ξ < 1$ for some $ξ \in (0, 1)$ , because $\cos$ is strictly decreasing on $[0, 2]$ and $\cos 0 = 1$ . Therefore $∣ J_{1} (1) ∣ = \sin 1 < 1 = ∣ J_{2} (1) ∣,$ so the inequality $∣ J_{1} (1) ∣ \geq ∣ J_{2} (1) ∣$ of the false claim is violated by the displayed pair, whose data satisfy every hypothesis of the claim. Hence the claim is false; the correct statement is the opposite comparison inequality of [F4], on its no-conjugate comparison interval. Both model fields are explicit, so the inherited ${A C}_{ω}$ of [A1] is not drawn on beyond its declaration.

Source locator

Datar §25.2 (printed p.185) states the lesson verbatim: "larger the curvature, smaller the Jacobi fields", and §25.3 proves the corresponding norm comparison; Eschenburg §3 (Rauch I, printed p.13) states the same direction. The refutation above uses the sphere-versus-Euclidean comparison pair, where the spherical field is $\sin t$ and the Euclidean field is $t$ .

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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)