In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is...
9 KB (1,164 words) - 03:07, 18 June 2024
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature...
59 KB (8,676 words) - 01:17, 20 August 2024
curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications. The curvature of a Riemannian manifold can be described...
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In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to...
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Riemann curvature tensor (redirect from Riemannian curvature)
sense made precise by the Jacobi equation. Let (M, g) be a Riemannian or pseudo-Riemannian manifold, and X ( M ) {\displaystyle {\mathfrak {X}}(M)} be the...
19 KB (2,904 words) - 22:21, 30 June 2024
of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of...
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Levi-Civita connection (category Riemannian geometry)
of a manifold (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry...
21 KB (3,370 words) - 12:50, 3 July 2024
Symmetric space (redirect from Locally Riemannian symmetric space)
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion...
45 KB (4,599 words) - 15:11, 4 July 2024
field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there...
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Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an...
13 KB (1,471 words) - 09:45, 15 August 2024
Ricci curvature (category Riemannian manifolds)
geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the...
34 KB (5,859 words) - 04:51, 6 July 2024
Geodesic (section Riemannian geometry)
in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization...
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List of formulas in Riemannian geometry Christoffel symbols Intrinsic metric Pseudo-Riemannian manifold Sub-Riemannian manifold Finsler geometry General...
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of such curves. Let M {\displaystyle M} be a (pseudo-)Riemannian manifold, g {\displaystyle g} the (pseudo-) metric tensor, γ : [ 0 , 1 ] → M {\displaystyle...
29 KB (5,174 words) - 00:27, 20 August 2024
Musical isomorphism (category Riemannian manifolds)
^{*}M} of a Riemannian or pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical...
9 KB (1,301 words) - 20:47, 17 July 2024
Isometry (redirect from Isometry (Riemannian geometry))
the manifold; a manifold with a (positive-definite) metric is a Riemannian manifold, one with an indefinite metric is a pseudo-Riemannian manifold. Thus...
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Riemannian manifold one can define notions of length, volume, and angle. Any smooth manifold can be given many different Riemannian metrics. A pseudo-Riemannian...
67 KB (9,495 words) - 02:22, 11 July 2024
geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the...
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Scalar curvature (category Riemannian geometry)
partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature...
35 KB (5,029 words) - 23:36, 30 May 2024
Conformal geometry (redirect from Conformal manifold)
spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up...
21 KB (3,351 words) - 09:08, 12 December 2023
Killing vector field (category Riemannian geometry)
named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the...
27 KB (4,720 words) - 21:59, 3 August 2024
In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, there...
4 KB (511 words) - 19:34, 4 July 2024
Line element (category Riemannian geometry)
line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of...
10 KB (1,564 words) - 01:51, 9 May 2024
A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In...
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trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations...
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Hermitian manifold Hyperkähler manifold Kähler manifold Lie group Pseudo-Riemannian manifold Riemannian manifold Sasakian manifold Spin manifold Symplectic...
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Volume form (redirect from Riemannian volume form)
Oriented pseudo-Riemannian manifolds have an associated canonical volume form. The following will only be about orientability of differentiable manifolds (it's...
14 KB (2,341 words) - 02:16, 8 May 2024
of the definitions given below. Connection Curvature Metric space Riemannian manifold See also: Glossary of general topology Glossary of differential geometry...
14 KB (2,090 words) - 01:41, 13 August 2024
Weyl tensor (redirect from Weyl manifold)
a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses...
10 KB (1,742 words) - 17:55, 29 January 2024
while relaxing this to a symmetric nondegenerate form yields a pseudo-Riemannian manifold. If V is finite-dimensional then, relative to some basis for V...
6 KB (850 words) - 02:14, 29 May 2024