Exponential map
# Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include:

* exponential map (Riemannian geometry) for a manifold with a Riemannian metric,
* exponential map (Lie theory) from a Lie algebra to a Lie group,
* More generally, in a manifold with an affine connection, $X\mapsto \gamma _{X}(1)$, where $\gamma _{X}$ is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections.
* Euler's formula forming the unit circle in the complex plane.
