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.