In Riemannian geometry, the exponential map is the map from (a subset of) the tangent space T_pM 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 name comes from the fact that it coincides with exponentiation of matrices in the case of bi-invariant metrics on Lie groups, when one is using a matrix representation of the group, and its Lie algebra as tangent space at the identity.

See also

• Campbell-Baker-Hausdorff formula