# Exponential map

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{{#invoke:Hatnote|hatnote}}
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, , where 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.