In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is...
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In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature...
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Riemann curvature tensor (redirect from Riemannian curvature)
Jacobi equation. Let ( M , g ) {\displaystyle (M,g)} be a Riemannian or pseudo-Riemannian manifold, and X ( M ) {\displaystyle {\mathfrak {X}}(M)} be the...
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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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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...
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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...
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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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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...
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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...
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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
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
In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, there...
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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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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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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 (1,303 words) - 18:39, 10 September 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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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
or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold, or more generally pseudo-Riemannian manifold, k-forms can...
34 KB (4,936 words) - 14:18, 10 September 2024
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
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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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
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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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...
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are Finsler manifolds if the norm of the vector space is smooth outside the origin. Riemannian manifolds (but not pseudo-Riemannian manifolds) are special...
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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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Orientability (redirect from Orientable manifold)
which gives the orientation character. A space-orientation of a pseudo-Riemannian manifold is identified with a section of the associated bundle O ( M...
25 KB (3,515 words) - 01:43, 10 August 2024
Splitting theorem (category Theorems in Riemannian geometry)
when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although...
9 KB (930 words) - 15:46, 22 July 2024