- pseudo-Riemannian manifold
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in differential geometry, a generalization of a Riemannian manifoldSyn: semi-Riemannian manifold
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pseudo-Riemannian manifold
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Pseudo-Riemannian manifold — In differential geometry, a pseudo Riemannian manifold (also called a semi Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann. The key difference between the two is that on a… … Wikipedia
Riemannian manifold — In Riemannian geometry, a Riemannian manifold ( M , g ) (with Riemannian metric g ) is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The… … Wikipedia
Riemannian geometry — Elliptic geometry is also sometimes called Riemannian geometry. Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent… … Wikipedia
Pseudo-Euclidean space — A pseudo Euclidean space is a finite dimensional real vector space together with a non degenerate indefinite quadratic form. Such a quadratic form can, after a change of coordinates, be written as : q(x) = left(x 1^2+cdots + x k^2 ight) left(x… … Wikipedia
Differentiable manifold — A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the middle chart the Tropic of Cancer is a smooth curve, whereas in the first it has a sharp… … Wikipedia
Curvature of Riemannian manifolds — In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous… … Wikipedia
Conformally flat manifold — A (pseudo )Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. More formally, let (M, g) be a pseudo Riemannian manifold. Then (M, g) is conformally flat if for… … Wikipedia
Fundamental theorem of Riemannian geometry — In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo Riemannian manifold) there is a unique torsion free metric connection, called the Levi Civita connection of the given metric … Wikipedia
Einstein manifold — In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo Riemannian manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to… … Wikipedia
Geodesic manifold — In mathematics, a geodesic manifold (or geodesically complete manifold) is a surface on which any two points can be joined by a shortest path, called a geodesic.DefinitionLet (M, g) be a (connected) (pseudo ) Riemannian manifold, and let gamma :… … Wikipedia
