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A Riemannian manifold (M, g) is said to be homogeneous if for every pair of points x and y in M, there is some isometry f of the Riemannian manifold sending x to y. This can be rephrased in the language of group actions as the requirement that the natural action of the isometry group is transitive.
There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. An example of this is the Clifton–Pohl torus . In the theory of general relativity , which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g. non-rotating uncharged ...
Klein quartic with 28 geodesics (marked by 7 colors and 4 patterns). In geometry, a geodesic (/ ˌ dʒ iː. ə ˈ d ɛ s ɪ k,-oʊ-,-ˈ d iː s ɪ k,-z ɪ k /) [1] [2] is a curve representing in some sense the locally [a] shortest [b] path between two points in a surface, or more generally in a Riemannian manifold.
Fix a point in a complete Riemannian manifold (,), and consider the tangent space.It is a standard result that for sufficiently small 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.
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry , leading to consequences in the theory of holonomy ; or algebraically through Lie theory , which ...
Solving the geodesic equations is a procedure used in mathematics, particularly Riemannian geometry, and in physics, particularly in general relativity, that results in obtaining geodesics. Physically, these represent the paths of (usually ideal) particles with no proper acceleration, their motion satisfying the geodesic equations.
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold.
Then the category of (,) manifolds is equivalent to the category of Riemannian manifolds which are locally isometric to (i.e. every point has a neighbourhood isometric to an open subset of ). Often the examples of X {\displaystyle X} are homogeneous under G {\displaystyle G} , for example one can take X = G {\displaystyle X=G} with a left ...