Relation between Geodesic and Exponential Map

Theorem

Let $\struct {M, g, \nabla}$ be a Riemannian or pseudo-Riemannian manifold without boundary endowed with the Levi-Civita connection.

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

Let $\exp$ be the exponential map.

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

Let $\gamma_v : I \to M$ be the unique maximal geodesic such that:

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

where $v \in TM$.

Then:

$\forall t \in I : \forall v \in TM : \map {\gamma_v} t = \map \exp {t v}$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. The Exponential Map