Definition:Restricted Exponential Map

Definition

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

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

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

Let $\EE \subseteq TM$ be the domain of the exponential map.

Let $\EE_p = \EE \cap T_p M$.

Then the restricted exponential map, denoted by $\exp_p$, is the mapping $\exp : \EE_p \to M$.

Further research is required in order to fill out the details. In particular: Actually, any connection in $TM$ will do, but source focuses on Levi-Civita You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

Sources

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