Properties of Domain of Exponential Map

Theorem

Let $\struct {M, g, \nabla}$ be a Riemannian or pseudo-Riemannian manifold endowed with the Levi-Civita connection.

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

Let $v \in T_p M$.

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 $\gamma'$ is the velocity of $\gamma$.

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

Let $\exp : \EE \to M$ be the exponential map where:

$\EE = \set {v \in TM : \text{$\gamma_v$ is defined on $I : \closedint 0 1 \subseteq I$}}$

Then $\EE$ is an open subset of $TM$ containing the image of the zero section.

Furthermore, $\forall p \in M$ the set $\EE_p \subseteq T_p M$ is star-shaped with respect to $0$.

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