---
tag: 275GQ9T7EB
kind: definition
name: Exponential map
msc: [53C22, 53B20]
defines: Y7Z0AGY1A7
uses: [7CBT4SM83Y, NSYMQMHPDN, WY6K3P4YCG]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry)
---

## Definition

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and $p \in M$.
For $v$ in the [[tangent space|WY6K3P4YCG]] $T_pM$, let $\gamma_v$ be the maximal [[geodesic|NSYMQMHPDN]] with $\gamma_v(0) = p$ and $\gamma_v'(0) = v$.
Let $\mathcal{E}_p \subseteq T_pM$ be the set of $v$ such that $\gamma_v$ is defined on an interval containing $[0, 1]$.
The *exponential map* at $p$ is
$$\exp_p \colon \mathcal{E}_p \to M, \qquad \exp_p(v) = \gamma_v(1).$$

## Remarks

Example: in $\R^n$ with the Euclidean metric, $\gamma_v(t) = p + tv$, so $\mathcal{E}_p = \R^n$ and $\exp_p(v) = p + v$.

Non-example: in the punctured plane $M = \R^2 \setminus \set{0}$ with $p = (1, 0)$ and $v = (-2, 0)$, the geodesic $\gamma_v(t) = (1 - 2t, 0)$ hits the removed origin at $t = 1/2$.
So $v \notin \mathcal{E}_p$, and $\exp_p(v)$ is not defined.

The set $\mathcal{E}_p$ is open and star-shaped about $0$, and $\exp_p$ is smooth on it.
By rescaling, $\gamma_v(t) = \exp_p(tv)$.
The differential of $\exp_p$ at $0$ is the identity of $T_pM$.
So, by the inverse function theorem, $\exp_p$ maps a neighbourhood of $0$ diffeomorphically onto a neighbourhood of $p$.