Existence and Uniqueness of Maximal Geodesic

Theorem

Let $M$ be a smooth manifold.

Let $TM$ be the tangent bundle of $M$.

Let $T_p M$ be the tangent space at $p \in M$.

Let $\nabla$ be a connection in $TM$.

Let $I \subseteq \R$ be an open real interval.

Let $\gamma : I \to M$ be a geodesic and $\gamma'$ its velocity.

Then $\forall p \in M$ and $\forall v \in T_p M$ there is a unique maximal geodesic $\gamma$ with:

$\map \gamma 0 = p$

$\map {\gamma'} 0 = v$

defined on some $I$ with $0 \in I$.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Geodesics