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Lower positive sectional curvature forces conjugate points

Statement

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $(M, g)$ be a complete Riemannian manifold of dimension $n \geq 2$ whose sectional curvatures are bounded below by $k > 0$ : every sectional curvature of every two-plane in every tangent space satisfies $K \geq k$ . Let $p \in M$ and let $γ : [0, \infty) \to M$ be a unit-speed geodesic with $γ (0) = p$ . Then $γ$ has a conjugate point to $p$ : the first conjugate instant $τ$ of $p$ along $γ$ is finite and $τ \leq \frac{π}{\sqrt{k}} .$

No strictness is asserted at the endpoint: $τ = π / \sqrt{k}$ is possible and is realized on the round sphere of curvature $k$ . Completeness is part of the hypotheses of the geometric setting of this page; the comparison argument below uses only the curvature bound and the absence of conjugate points.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1], the complete $n$ -dimensional Riemannian manifold $(M, g)$ with $K \geq k > 0$ , a point $p \in M$ and a unit-speed geodesic $γ : [0, \infty) \to M$ from $p$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried through the curvature and index-form suppliers used below; the Jacobi and parallel initial-value constructions require no choice.

[F1]

Rauch comparison, first form (Rauch comparison theorem first form): for unit-speed geodesics in $n$ -dimensional manifolds and normal Jacobi fields $J_{1}, J_{2}$ with $J_{i} (0) = 0$ and equal positive initial-derivative norms, if every relevant radial sectional curvature of $M_{1}$ is at least every such curvature of $M_{2}$ , and $M_{1}$ has no conjugate point in $(0, T]$ , then $∣ J_{1} (t) ∣ \leq ∣ J_{2} (t) ∣$ on $[0, T]$ .

[F2]

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

[F3]

Constant-curvature model (Constant sectional curvature and space form, Curvature tensor of constant sectional curvature): in a manifold of constant sectional curvature $k$ the curvature tensor is $R (X, Y) Z = k (g (Y, Z) X - g (X, Z) Y)$ , so in particular every radial sectional curvature equals $k$ and $R (X, \dot{γ}) \dot{γ} = k (X - g (X, \dot{γ}) \dot{γ})$ .

[F4]

Parallel transport (Existence and uniqueness of parallel sections, Levi civita parallel transport preserves lengths angles and volume): along a geodesic there is a unique smooth parallel field with prescribed value at one time, and parallel transport is an isometry preserving orthogonality.

[F5]

Radial Jacobi data (Radial Jacobi tensor, Radial jacobi tensor is invertible before the first conjugate point, Jacobi field): for $w$ in the normal space at $γ (0)$ the radial field $J (t) = A (t) w$ is the unique normal Jacobi field with $J (0) = 0$ , $D_{t} J (0) = w$ , and $A (t)$ is an isomorphism of normal spaces for every $t > 0$ before the first conjugate instant; a Jacobi field with $J (t_{0}) = 0 = D_{t} J (t_{0})$ is identically zero.

[F6]

Conjugate points (Conjugate points along a geodesic and their multiplicity): $γ (t_{0})$ , $t_{0} > 0$ , is conjugate to $γ (0)$ along $γ$ exactly when there is a nonzero Jacobi field along $γ ∣_{[0, t_{0}]}$ vanishing at both ends; the first conjugate instant is the infimum of such $t_{0}$ .

[F7]

Sectional curvature (Sectional curvature): a plane containing $\dot{γ} (t)$ has sectional curvature $\sec$ , and $K \geq k$ means $\sec \geq k$ for every plane.

Proof

technique · direct: identify the explicit radial Jacobi field of the constant-curvature model, whose norm is $\operatorname{sn}_k$ and which vanishes at $\pi/\sqrt k$, and compare any radial field of $M$ with it through Rauch's first form; the model's zero forces a conjugate point

1.1F2F3F4given

The model radial field. [F2, F3, F4, given] Let $M_{k}^{n}$ be the complete, simply connected space form of constant curvature $k$ , let $\tilde{γ}$ be a unit-speed geodesic in $M_{k}^{n}$ and let $w \in T_{\tilde{γ} (0)} M_{k}^{n}$ satisfy $∣ w ∣ = 1$ and $w ⊥ {\tilde{γ}}^{'} (0)$ ; write $Φ_{t}$ for parallel transport along $\tilde{γ}$ and $J_{k} (t) : = {sn}_{k} (t) Φ_{t} w .$ Then $D_{t} J_{k} = {sn}_{k}^{'} Φ_{t} w$ and $D_{t}^{2} J_{k} = {sn}_{k}^{''} Φ_{t} w = - k {sn}_{k} Φ_{t} w$ , while by [F3] and the normality of $J_{k}$ (preserved by parallel transport [F4]) $R (J_{k}, {\tilde{γ}}^{'}) {\tilde{γ}}^{'} = k (g ({\tilde{γ}}^{'}, {\tilde{γ}}^{'}) J_{k} - g (J_{k}, {\tilde{γ}}^{'}) {\tilde{γ}}^{'}) = k J_{k} .$ Hence $D_{t}^{2} J_{k} + R (J_{k}, {\tilde{γ}}^{'}) {\tilde{γ}}^{'} = 0$ : the field is a normal Jacobi field with $J_{k} (0) = 0, D_{t} J_{k} (0) = w, ∣ J_{k} (t) ∣ = {sn}_{k} (t) (0 \leq t \leq π / \sqrt{k}),$ and by [F2] it satisfies $J_{k} (t) \neq 0$ for $0 < t < π / \sqrt{k}$ and $J_{k} (π / \sqrt{k}) = 0$ .

2.1F1F5F6F7step 1.1given∎

Rauch comparison and the conjugate point. [F1, F5, F6, F7, step 1.1, given] Let $T_{k} : = π / \sqrt{k}$ and suppose, for contradiction, that no $t \in (0, T_{k}]$ is a conjugate instant of $p$ along $γ$ . Choose a unit normal vector $w_{1} \in {\dot{γ} (0)}^{⊥}$ , which exists because $n \geq 2$ , and let $J_{1} (t) : = A (t) w_{1}$ be the radial Jacobi field with $J_{1} (0) = 0$ and $∣ D_{t} J_{1} (0) ∣ = 1$ [F5]. By the supposition and [F5], $A (t)$ is invertible for $0 < t \leq T_{k}$ , so $J_{1} (t) \neq 0$ there; in particular $D_{t} J_{1} (0) = w_{1} \neq 0$ , so $J_{1}$ is not the zero field [F5]. Apply Rauch's first form [F1] with $M_{1} : = M, γ_{1} : = γ, J_{1} and M_{2} : = M_{k}^{n}, γ_{2} : = \tilde{γ}, J_{2} : = J_{k},$ the model field of step 1.1: both fields vanish at $0$ and have initial derivative norm $1$ . Hypothesis 1 of [F1] holds because every radial sectional curvature of $M$ is at least $k$ by [F7] while every radial sectional curvature of the model equals $k$ by [F3]; hypothesis 2 holds by the contradiction supposition. The conclusion is $∣ J_{1} (T_{k}) ∣ \leq ∣ J_{2} (T_{k}) ∣ = ∣ {sn}_{k} (T_{k}) ∣ = 0,$ so $J_{1} (T_{k}) = 0$ . Thus the nonzero Jacobi field $J_{1}$ vanishes at $t = 0$ and $t = T_{k}$ , that is, $T_{k}$ is a conjugate instant of $p$ along $γ$ [F6], contradicting the supposition. Therefore some $t \in (0, T_{k}]$ is a conjugate instant, and the first conjugate instant $τ$ satisfies $τ \leq T_{k}$ . The proof uses only the hypothetical field and the single model field fixed in step 1.1, so the inherited ${A C}_{ω}$ of [A1] is not drawn on beyond its declaration.

Source locator

Datar Corollary 25.3.3 with its proof (printed p.189) obtains the conjugate point bound $\leq π / \sqrt{κ}$ under $\sec \geq κ > 0$ exactly as the contrapositive argument above: if no conjugate point occurred up to $π / \sqrt{κ}$ , the comparison with the spherical radial field ${sn}_{κ}$ would force the actual field to vanish there. Eschenburg §3 (printed p.13) contains the same content as the strict specialisation of Rauch I; the proof here uses the in-run Rauch first form and the published model-function and space-form suppliers.

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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)