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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.
There are one-to-one correspondences {one-parameter subgroups of G} ⇔ {left-invariant vector fields on G} ⇔ g = T e G. Let G be a Lie group and g its Lie algebra. The exponential map is a map exp : g → G given by exp(X) = γ(1) where γ is the integral curve starting at the identity in G generated by X. The exponential map is smooth.
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 ...
The ordinary exponential function of mathematical analysis is a special case of the exponential map when is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however ...
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.
Parameter plane of the complex exponential family f(z)=exp(z)+c with 8 external ( parameter) rays. In the theory of dynamical systems, the exponential map can be used as the evolution function of the discrete nonlinear dynamical system. [1]
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 formula for the exponential results from reducing the powers of G in the series expansion and identifying the respective series coefficients of G 2 and G with −cos(θ) and sin(θ) respectively. The second expression here for e Gθ is the same as the expression for R ( θ ) in the article containing the derivation of the generator , R ( θ ...