Exponential map

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

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

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.

Exponential map - Your Art History Reference Guide!

See also