# Exponential mapping

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A mapping of the tangent space of a manifold into . It is defined by a connection given on and is a far-reaching generalization of the ordinary exponential function regarded as a mapping of a straight line into itself.
1) Let be a -manifold with an affine connection, let be a point in , let be the tangent space to at , let be a non-zero vector in , and let be the geodesic passing through in the direction of . There is an open neighbourhood of the point in and an open neighbourhood of in such that the mapping is a diffeomorphism of onto . This mapping is called the exponential mapping at and is denoted by . A neighbourhood is called normal if: 1) the mapping maps onto diffeomorphically; and 2) and imply that . In this case is said to be a normal neighbourhood of the point in the manifold . Every has a convex normal neighbourhood : Any two points of such a neighbourhood can be joined by exactly one geodesic segment lying in . If is a complete Riemannian manifold, then is a surjective mapping of onto .
2) Let be a Lie group with identity and let be the corresponding Lie algebra consisting of the tangent vectors to at . For every vector there is a unique differentiable homomorphism of the group into such that the tangent vector to at coincides with . The mapping is called the exponential mapping of the algebra into the group . There is an open neighbourhood of the point in and an open neighbourhood of in such that is a diffeomorphism of onto . Let be some basis for the algebra . The mapping is a coordinate system on ; these coordinates are called canonical.
The concept of an exponential mapping of a Lie group can also be approached from another point of view. There is a one-to-one correspondence between the set of all affine connections on that are invariant relative to the group of left translations and the set of bilinear functions . It turns out that the exponential mapping of the algebra into the group coincides with the mapping of the tangent space of into the manifold at the point in this manifold with respect to the left-invariant affine connection corresponding to any skew-symmetric bilinear function .