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Exponential map (Riemannian geometry)

This article is about the exponential map in differential geometry. For discrete dynamical systems, see Exponential map (discrete dynamical systems). For the exponential map from a Lie algebra to a Lie group, see Exponential map (Lie theory).
The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography.

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.

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