exponential map

{{about|the exponential map in differential geometry|discrete dynamical systems|Exponential map (discrete dynamical systems)}}

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, $X\; \backslash mapsto\; \backslash gamma\_X(1)$, where $\backslash gamma\_X$ 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.

{{disambig}}