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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 ...
The exponential map is a mapping from the tangent space at p to M: : which is a diffeomorphism in a neighborhood of zero. 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.
Exponential function; Exponential generating function; Exponential-Golomb coding; Exponential growth; Exponential hierarchy; Exponential integral; Exponential integrator; Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation; Exponential object (category theory ...
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]
Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for e A [13] For any skew-symmetric matrix A, exp(A) is always a rotation matrix. [nb 3] An important practical example is the 3 × 3 case.
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 ...
The exponential map is smooth. For a fixed X, the map t ↦ exp(tX) is the one-parameter subgroup of G generated by X. The exponential map restricts to a diffeomorphism from some neighborhood of 0 in g to a neighborhood of e in G. The image of the exponential map always lies in the connected component of the identity in G.