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.

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