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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 ...
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 ...
where for every direction in the base space, S n, the fiber over it in the total space, SO(n + 1), is a copy of the fiber space, SO(n), namely the rotations that keep that direction fixed. Thus we can build an n × n rotation matrix by starting with a 2 × 2 matrix, aiming its fixed axis on S 2 (the ordinary sphere in three-dimensional space ...
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 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.
That this gives a one-parameter subgroup follows directly from properties of the exponential map. [12] The exponential map provides a diffeomorphism between a neighborhood of the origin in the 𝖘𝖔(3) and a neighborhood of the identity in the SO(3). [13] For a proof, see Closed subgroup theorem. The exponential map is surjective.
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.