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

For <math>v\in T_p M<math> there is a unique geodesic <math>\gamma^{}_v<math> such that <math>\gamma^{}_{}(0)=p<math> having a tangent vector <math>\gamma'(0)=v_{}^{}<math>. Then <math>exp_p(v)=\gamma_v^{}(1).<math>

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