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Cut time does not exceed first conjugate time

Statement

Assume exactly the inherited Axiom of Countable Choice ${A C}_{ω}$ , carried through the declared dependencies. Let $(M, g)$ be a complete, connected, boundaryless, finite-dimensional Riemannian manifold, let $p \in M$ , let $v \in S_{p} M$ be a unit tangent vector, write $γ (t) = \exp_{p} (t v)$ for the radial geodesic and let $c : = c_{p} (v) \in (0, + \infty]$ be its cut time.

(a) If $t > 0$ and $γ (0) = p$ and $γ (t)$ are conjugate along $γ ∣_{[0, t]}$ , then $c \leq t$ .

(b) Consequently, if the set $C : = {t > 0 : γ (0) and γ (t) are conjugate along γ ∣_{[0, t]}}$ of positive conjugate instants is nonempty, then $c \leq \inf C$ ; the infimum $\inf C$ is the first conjugate time of the geodesic $γ$ , and $c = + \infty$ is possible only when $C = \emptyset$ .

The corollary asserts the inequality $c \leq \inf C$ only; it does not assert that $\inf C$ is attained, that is, that a first conjugate instant exists. Dimension zero has no instance, since there is no unit tangent vector.

Facts & Assumptions

Given: The complete connected boundaryless Riemannian manifold, the point $p$ , the unit vector $v$ , the radial geodesic $γ (t) = \exp_{p} (t v)$ , its cut time $c = c_{p} (v)$ and the set $C$ of positive conjugate instants.

[A1]

Countable choice is the assumption ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ), inherited exactly through the declared Hopf-Rinow, cut-time and conjugacy interfaces. No full Axiom of Choice is used.

[F1]

The cut time is $c_{p} (v) = \sup {t > 0 : d_{g} (p, \exp_{p} (t v)) = t} \in (0, + \infty]$ , and $S_{p} M = {w \in T_{p} M : ∣ w ∣_{g} = 1}$ (Cut time in a unit tangent direction).

[F2]

Converse for a conjugate instant. If $t > 0$ and $γ (0)$ , $γ (t)$ are conjugate along $γ ∣_{[0, t]}$ , then $c \leq t$ (Characterization of a cut point, clause (b)(1)).

[F3]

The points $γ (a)$ and $γ (b)$ are conjugate along an affinely parametrized geodesic segment exactly when some nonzero Jacobi field along it vanishes at both endpoints (Conjugate points along a geodesic and their multiplicity). In particular conjugacy pertains to the specified geodesic segment $γ ∣_{[0, t]}$ .

[F4]

If $S \subseteq R$ is nonempty and bounded below, then $\inf S$ exists, and it is a greatest lower bound: it is a lower bound of $S$ , and $ℓ^{'} \leq \inf S$ for every lower bound $ℓ^{'}$ of $S$ (Every nonempty set bounded below has an infimum, Greatest lower bound (infimum)).

Proof

technique · apply the converse direction of the cut-point characterization to each positive conjugate instant, then take the infimum

1.1F1F2F3given

Let $t > 0$ and suppose that $γ (0)$ and $γ (t)$ are conjugate along $γ ∣_{[0, t]}$ . By [F3] this is conjugacy along the specified radial segment, so clause (b)(1) of the characterization [F2] applies and gives $c \leq t$ . Since $t < + \infty$ and the order on $(0, + \infty]$ is the usual one, whenever $C$ is nonempty this exhibits $c < + \infty$ : the cut time is finite as soon as some positive conjugate instant exists.

2.1F4givenstep 1.1

If $C = \emptyset$ , statement (b) asserts nothing and there is nothing to prove; $c$ may be finite or $+ \infty$ in that case. Suppose instead that $C \neq \emptyset$ . Since $C \subseteq (0, \infty)$ , the number $0$ is a lower bound of $C$ and $C$ is nonempty and bounded below, so [F4] provides $\inf C$ . By step 1.1 every $t \in C$ satisfies $t \geq c$ , that is, $c$ is a lower bound of $C$ ; the greatest-lower-bound property in [F4] then gives $c \leq \inf C$ .

3.1A1F1F3F4step 1.1step 2.1

Boundary and choice audit. The positive time constraint $t > 0$ excludes the degenerate instant $0$ , which lies outside the scope of the conjugacy definition [F3]: that definition is stated for a nondegenerate segment $a < b$ , and a constant geodesic has no conjugate pairs at all. In dimension zero there is no unit tangent vector, so there is no instance; in dimension one the unit sphere $S_{p} M$ consists of the two unit vectors and the argument applies verbatim, while the radial geodesic is nonconstant because $∣ v ∣_{g} = 1$ by [F1]. The empty manifold has no point $p$ . When $C \neq \emptyset$ the set $C$ is a nonempty subset of $(0, \infty)$ and its infimum is a finite real number, while $c$ is then finite by step 1.1. No attainment of $\inf C$ is asserted: the corollary proves only the inequality, which is exactly the content of the statement. Exactly the inherited ${A C}_{ω}$ of [A1] is used, through the cut-time and characterization interfaces; the infimum of a nonempty bounded-below set of reals is supplied by [F4] and involves no choice.

$□$

Source locator

Lee, Riemannian Manifolds: An Introduction to Curvature, Chapter 10, printed pp.173-190, states the cut-point criterion and that the cut point occurs at or before the first conjugate point; Datar, Lectures on Riemannian Geometry, Lemma 23.2.2 and proof, printed pp.167-169, supplies the converse direction of the characterization that is applied here at each conjugate instant. The infimum formulation and its boundary cases are carried out above; nothing is quoted.

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Sources

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997) (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (2025) (standard reference, not scraped)