In Riemannian geometry, the exponential map is the map from (a subset of) the tangent space TpM 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