Geodesic flow
In differential geometry, the exponential map is the map from a subset of the tangent space  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 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.
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