In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is...

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature...

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...

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...

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...

of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of...

of a manifold (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry...

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an...

In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion...

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...

field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there...

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...

List of formulas in Riemannian geometry Christoffel symbols Intrinsic metric Pseudo-Riemannian manifold Sub-Riemannian manifold Finsler geometry General...

^{*}M} of a Riemannian or pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical...

of such curves. Let M {\displaystyle M} be a (pseudo-)Riemannian manifold, g {\displaystyle g} the (pseudo-) metric tensor, γ : [ 0 , 1 ] → M {\displaystyle...

Riemannian manifold one can define notions of length, volume, and angle. Any smooth manifold can be given many different Riemannian metrics. A pseudo-Riemannian...

geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the...

the manifold; a manifold with a (positive-definite) metric is a Riemannian manifold, one with an indefinite metric is a pseudo-Riemannian manifold. Thus...

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...

named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the...

partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature...

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, there...

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...

Hermitian manifold Hyperkähler manifold Kähler manifold Lie group Pseudo-Riemannian manifold Riemannian manifold Sasakian manifold Spin manifold Symplectic...

Oriented pseudo-Riemannian manifolds have an associated canonical volume form. The following will only be about orientability of differentiable manifolds (it's...

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...

of the definitions given below. Connection Curvature Metric space Riemannian manifold See also: Glossary of general topology Glossary of differential geometry...

a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses...

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...

are Finsler manifolds if the norm of the vector space is smooth outside the origin. Riemannian manifolds (but not pseudo-Riemannian manifolds) are special...