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The metric tensor g on a smooth manifold M=(M,g) is said to be semi-Riemannian if the index of g is nonzero. In nearly all literature, the term semi-Riemannian is used ...

A weak pseudo-Riemannian metric on a smooth manifold M is a (0,2) tensor field g which is symmetric and for which, at each m in M, g_m(v_m,w_m)=0 for all w_m in T_mM implies ...

The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open ...

Differential geometry is the study of Riemannian manifolds. Differential geometry deals with metrical notions on manifolds, while differential topology deals with nonmetrical ...

In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid's postulates, but each uses its own version of the ...

Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no ...

Any n-dimensional Riemannian manifold can be locally embedded into an (n+1)-dimensional manifold with Ricci curvature Tensor R_(ab)=0. A similar version of the theorem for a ...

A metric space X is isometric to a metric space Y if there is a bijection f between X and Y that preserves distances. That is, d(a,b)=d(f(a),f(b)). In the context of ...

A geometry constructed without reference to measurement. The only primitive concepts are those of points and intermediacy. There are 10 axioms underlying ordered geometry.