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exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine connection, (), where is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections.
In these important special cases, the exponential map is known to always be surjective: G is connected and compact, [5] G is connected and nilpotent (for example, G connected and abelian), or = (). [6] For groups not satisfying any of the above conditions, the exponential map may or may not be surjective.
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 T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical ...
It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n×n real or complex matrix. The exponential of X, denoted by e X or exp(X), is the n×n matrix given by the power series = =!
The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective.
The exponential map is defined by exp p (v) = c v (1) and gives a diffeomorphism between a disc ‖v‖ < δ and a neighbourhood of p; more generally the map sending (p, v) to exp p (v) gives a local diffeomorphism onto a neighbourhood of (p, p). The exponential map gives geodesic normal coordinates near p. [63]
The maps π and Π are Lie algebra and group representations respectively, and exp is the exponential mapping. The diagram commutes only up to a sign if Π is projective. We now outline the proof of the main results above.
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds .