Exponential Map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection. Two important special cases of this are the exponential map for a manifold with a Riemannian metric, and the exponential map from a Lie algebra to a Lie group.

Read more about Exponential MapDefinition, Lie Theory, Riemannian Geometry, Relationships

Other articles related to "exponential map, exponential maps, exponential, map":

Gauss's Lemma (Riemannian Geometry)
... The exponential map is a mapping from the tangent space at p to M which is a diffeomorphism in a neighborhood of zero ... small radius in TpM under the exponential map is perpendicular to all geodesics originating at p ... The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and normal ...
Exponential Map - Relationships
... metric—a pseudo-Riemannian metric invariant under both left and right translation—the exponential maps of the pseudo-Riemannian structure are the same as the exponential maps of ... Take the example that gives the "honest" exponential map ...
Skew-symmetric Matrix - Infinitesimal Rotations
... of two skew-symmetric matrices is again skew-symmetric The matrix exponential of a skew-symmetric matrix A is then an orthogonal matrix R The image of the exponential map ... Moreover, since the exponential map of a connected compact Lie group is always surjective, it turns out that every orthogonal matrix with unit ... important case of dimension n=2, the exponential representation for an orthogonal matrix reduces to the well-known polar form of a complex number of unit modulus ...
Cut Locus (Riemannian Manifold) - Definition
... in, the curve defined by the Riemannian exponential map, for belonging to the interval is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints ... Here denotes the exponential map from ... of in is defined to be image of the cut locus of in the tangent space under the exponential map at ...
Cartan–Hadamard Theorem - Riemannian Geometry
... In fact, for complete manifolds on non-positive curvature the exponential map based at any point of the manifold is a covering map ... also for Hilbert manifolds in the sense that the exponential map of a non-positively curved geodesically complete connected manifold is a covering map (McAlpin 1965 Lang 1991, IX, §3) ... Completeness here is understood in the sense that the exponential map is defined on the whole tangent space of a point ...

Famous quotes containing the word map:

    When I had mapped the pond ... I laid a rule on the map lengthwise, and then breadthwise, and found, to my surprise, that the line of greatest length intersected the line of greatest breadth exactly at the point of greatest depth.
    Henry David Thoreau (1817–1862)