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Every Riemannian symmetric space is homogeneous, and consequently is geodesically complete and has constant scalar curvature. However, Riemannian symmetric spaces also have a much stronger curvature property not possessed by most homogeneous Riemannian manifolds, namely that the Riemann curvature tensor and Ricci curvature are parallel.
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.
The existence of isothermal coordinates on a smooth two-dimensional Riemannian manifold is a corollary of the standard local solvability result in the analysis of elliptic partial differential equations. In the present context, the relevant elliptic equation is the condition for a function to be harmonic relative to
In geodesic coordinates, it is easy to check that the geodesics through zero minimize length. The topology on the Riemannian manifold is then given by a distance function d(p,q), namely the infimum of the lengths of piecewise smooth paths between p and q. This distance is realised locally by geodesics, so that in normal coordinates d(0,v ...
In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M be a Riemannian manifold, equipped with its Levi-Civita connection, and p a point of M. The exponential map is a mapping from the tangent space ...
Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow , who published it in 1931. [ 1 ] Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem to the context of certain types of metric spaces .
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 ...
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 affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.