Exponential map

Builds on Riemannian manifold Geodesic Tangent bundle

Definition

Let $(M, g)$ be a Riemannian manifold and $p \in M$ . For $v$ in the tangent space $T_pM$ , let $\gamma_v$ be the maximal geodesic 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 $p$ .

Reading path

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Geodesic concept
Riemannian manifold concept
Tangent bundle concept
Exponential map definition

Tag	`275GQ9T7EB`
URI	`m2://275GQ9T7EB`
Kind	definition
Writing	W0 stub
MSC	53C22, 53B20
Defines	Exponential map
Uses	Riemannian manifold, Geodesic, Tangent bundle
Changed	2 November 2026

Exponential map

Definition

Remarks

Reading path

Further reading

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