DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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Geodesic spray

Definition

Assume countable choice ${A C}_{ω}$ . For an affine connection on $M$ , consider in every induced tangent-bundle chart $(x^{i}, v^{i})$ the local formula $S = v^{i} \frac{\partial}{\partial x^{i}} - Γ^{k}_{i j} (x) v^{i} v^{j} \frac{\partial}{\partial v^{k}} .$ The geodesic spray is the smooth vector field on $T M$ obtained from these chartwise formulas. Their overlap agreement, and hence the existence and uniqueness of this global vector field, is proved in The geodesic spray is a well-defined smooth vector field on TM ↗.

Facts & Assumptions

Given: The affine connection and an induced tangent-bundle chart.

[F1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is ${A C}_{ω}$ .

[F2]

Under [F1], Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure makes $T M$ a smooth manifold with the charts of The induced tangent bundle chart.

[F3]

Coordinate geodesic equation rewrites the second-order geodesic equation as ${\dot{x}}^{k} = v^{k}$ and ${\dot{v}}^{k} = - Γ^{k}_{i j} v^{i} v^{j}$ .

Verification

1.1F2F3given

In the $2 n$ coordinates $(x, v)$ supplied by [F2], the displayed expression has base components $v^{i}$ and fibre components $- Γ^{k}_{i j} (x) v^{i} v^{j}$ . The Christoffel functions are smooth, so every component is smooth. An integral curve of this local expression obeys exactly the first-order system in [F3].

2.1F1F2F3step 1.1∎

At a zero vector $v = 0$ both component lists vanish, so the zero section consists of stationary points of the local spray. For $n = 1$ the formula is $v \partial_{x} - Γ^{1}_{11} (x) v^{2} \partial_{v}$ ; for $n = 0$ it is the zero vector field on the discrete zero section. Empty $M$ gives empty $T M$ . No claim of overlap agreement is used here; that is the next lemma. The only choice principle is the explicitly assumed ${A C}_{ω}$ in [F1]–[F2], used to obtain the global smooth-manifold structure on $T M$ ; the coordinate formula itself is choice-free.

Depends on

Dependency tree · 2 levels

Sources

Ved Datar, Lectures on Riemannian Geometry, proof of Theorem 15.2.1, pp.115–117