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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.
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 ...
Gauss' lemma asserts that the image of a sphere of sufficiently small radius in T p M under the exponential map is perpendicular to all geodesics originating at p. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and normal coordinates.
The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G {\displaystyle G} is the multiplicative group of positive real numbers (whose Lie algebra is the additive group ...
The standard definition of ordinal exponentiation with base α is: =, =, when has an immediate predecessor . = {< <}, whenever is a limit ordinal. From this definition, it follows that for any fixed ordinal α > 1, the mapping is a normal function, so it has arbitrarily large fixed points by the fixed-point lemma for normal functions.
In case G is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted exp:g → G, is analytic and has as such a derivative d / dt exp(X(t)):Tg → TG, where X(t) is a C 1 path in the Lie algebra, and a closely related differential dexp:Tg → TG. [2] The formula for dexp was first proved ...
The second one can be mapped to the first using the fact that . = + (), so : + is the same under the transformation = + (). The only difference is that, due to multi-valued properties of exponentiation, there may be a few select cases that can only be found in one version.
In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3). This formula is variously credited to Leonhard Euler, Olinde Rodrigues, or a combination of the two. A detailed historical analysis in 1989 concluded that the formula should be attributed to ...