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Exponential map - Riemannian geometry
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Exponential map, Exponential map - Definition, Exponential map - Lie theory, Exponential map - Properties, Exponential map - Relationships, Exponential map - Riemannian geometry, Exponential function, Matrix exponential, List of exponential topics

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Exponential map - Riemannian geometry: Encyclopedia II - Exponential map - Riemannian geometry

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. Exponential map - Definition. For v ∈ TpM, there is a unique geodesic γv satisfying γv(0) = p such that the tangent vector γ′v(0) = v. Then the corresponding exponential map is defined by expp(v) = ...

The exponential map is a fundamental construction in the theory of Lie groups. It is a map from the Lie algebra of a Lie group to the group which allows one to completely recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras. The ordinary exponential function of mathematical analysis may be viewed as a special case of the exponential map when G is the multiplicative group of positive real ...

The two notions of the exponential map coincide in the case of Lie groups equipped with bi-invariant metrics (i.e. Riemannian metrics invariant under left and right translation). In this case the geodesics through the identity are precisely the one-parameter subgroups of G. Take the example that gives the "honest" exponential map. Consider the positive real numbers R+, a Lie group under the usual multiplication. Then each tangent space is just R. On each copy of R at the point y, we introduce the modified inner product <u,v>y< ...

Exponential map - Riemannian geometry

A Wisdom Archive on Exponential map - Riemannian geometry

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