Exponential map

In differential geometry, the exponential map is the map from a subset of the tangent space of a Riemannian manifold M to M itself. It is defined in the following way:

For there is a unique geodesic such that having a tangent vector . Then

The geodesic flow is the corresponding flow on the tangent bundle TM of M. Its trajectories are of the form where is a geodesic.

The name comes from the fact that it coincides with exponentiation of matrices in the case of certain metrics on Lie groups, when one is using a matrix representation of the group, and its Lie algebra as tangent space at the identity.