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Rauch comparison between euclidean and spherical geodesics

Example

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

Facts & Assumptions

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

[A1]

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

[F1]

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

[F2]

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

[F3]

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

[F4]

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

[F5]

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

Verification

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

1.1F1F2F3given

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

2.1F1F2F4step 1.1given

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

3.1F5step 2.1given∎

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

Source locator

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

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Sources

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