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  2. Exponential map - Wikipedia

    en.wikipedia.org/wiki/Exponential_map

    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.

  3. 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 ...

  4. 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.

  5. Vector flow - Wikipedia

    en.wikipedia.org/wiki/Vector_flow

    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.

  6. 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 ...

  7. 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 ...

  8. List of exponential topics - Wikipedia

    en.wikipedia.org/wiki/List_of_exponential_topics

    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) Exponential polynomials ...

  9. Exponential map (discrete dynamical systems) - Wikipedia

    en.wikipedia.org/wiki/Exponential_map_(discrete...

    There are many forms of these maps, [2] many of which are equivalent under a coordinate transformation. For example two of the most common ones are: : +: The second one can be mapped to the first using the fact that . = + (), so : + is the same under the transformation = + ().