Exponential map explained

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.