In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, 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)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of ...
from a Lie algebra to a Lie group,
* More generally, in a manifold with an
affine connection,
, where
is a
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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.
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