---
tag: Y7Z0AGY1A7
kind: concept
name: Exponential map
msc: [53C22, 53B20]
sources:
  - url: https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry)
same_as:
  wikidata: Q1384674
  wikipedia: Exponential map (Riemannian geometry)
---

## Summary

Let $p$ be a point of a [[Riemannian manifold|7CBT4SM83Y]].
The exponential map at $p$ sends $v \in T_pM$ to the point reached at time $1$ by the [[geodesic|NSYMQMHPDN]] from $p$ with velocity $v$.
It maps a small ball around $0$ in the tangent space diffeomorphically onto a neighbourhood of $p$.
So it gives coordinates adapted to the geometry, the [[normal coordinates|RRHKMGNKG9]].
It is central to the study of completeness, conjugate points and the cut locus.
For a bi-invariant metric on a Lie group, it agrees at the identity with the exponential map of the Lie group.

## Explanations

### L1 human

Stand at a point $p$, choose a direction and a distance, and walk straight ahead, along a geodesic, for that distance.
The point where you stop is the exponential map of the vector with that direction and length.
On the unit sphere, starting at the north pole, the vectors of length $\pi/2$ go to the equator and all vectors of length $\pi$ go to the south pole.
This is like wrapping the flat tangent plane around the sphere so that straight lines through the origin become great circles through $p$.