Exponential map
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In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold M to M itself.
Definition
For v ∈ TpM, there is a unique geodesic γv satisfying γ(0) = p such that the tangent vector γ′(0) = v. Then the corresponding exponential map is defined by expp(v) = γv(1).
Why "exponential"?
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.