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 $\exp : \EE_p \to M$.