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In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include: exponential map (Riemannian geometry) for a manifold with a Riemannian metric, exponential map (Lie theory) from a Lie algebra to a Lie group,
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 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 formula is named after Henry Frederick Baker, John Edward Campbell, ... = XY − YX; the exponential map is the standard exponential map of matrices ...
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 ...
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 ...
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]
Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for e A [14] For any skew-symmetric matrix A, exp(A) is always a rotation matrix. [nb 3] An important practical example is the 3 × 3 case.