# Definition: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 $v \in T_p M$.

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 $\EE \subseteq TM$ be the set defined as:

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

Then the exponential map, denoted by $\exp$, is the mapping $\exp : \EE \to M$ such that:

$\map \exp v = \map {\gamma_v} 1$

## Also see

• Results about exponential maps can be found here.