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  2. Exponential map - Wikipedia

    en.wikipedia.org/wiki/Exponential_map

    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,

  3. Exponential map (Lie theory) - Wikipedia

    en.wikipedia.org/wiki/Exponential_map_(Lie_theory)

    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 ...

  4. Exponential map (Riemannian geometry) - Wikipedia

    en.wikipedia.org/wiki/Exponential_map...

    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 ...

  5. Baker–Campbell–Hausdorff formula - Wikipedia

    en.wikipedia.org/wiki/Baker–Campbell...

    The formula is named after Henry Frederick Baker, John Edward Campbell, ... = XY − YX; the exponential map is the standard exponential map of matrices ...

  6. Derivative of the exponential map - Wikipedia

    en.wikipedia.org/wiki/Derivative_of_the...

    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 ...

  7. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.

  8. Rodrigues' rotation formula - Wikipedia

    en.wikipedia.org/wiki/Rodrigues'_rotation_formula

    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 ...

  9. Gauss's lemma (Riemannian geometry) - Wikipedia

    en.wikipedia.org/wiki/Gauss's_lemma_(Riemannian...

    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.