field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the...

Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from...

the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated Riemann curvature tensor. In part, this...

given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions...

the full Riemann curvature tensor. Alternatively, in a coordinate-free notation one may use Riem for the Riemann tensor, Ric for the Ricci tensor and R for...

(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvature tensor, ...), and...

differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can...

A a tensor field on M. Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor field...

metric (and the associated curvature tensors) to the stress–energy tensor T μ ν {\displaystyle T_{\mu \nu }} . This tensor equation is a complicated set...

term curvature tensor may refer to: the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds; the curvature of...

way of measuring the curvature of a manifold is with an object called the Riemann (curvature) tensor. This tensor measures curvature by use of an affine...

the Ricci tensor. The Riemann curvature tensor can be expressed in terms of the covariant derivative. The Einstein tensor G is a rank-2 tensor defined over...

geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian...

Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar curvature is the unique metric contraction of the Ricci tensor. One...

notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern...

mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature. The principal symbol of the map g ↦ Rm g {\displaystyle...

"identity". The Ricci and Bianchi identities given in terms of the Riemann curvature tensor illustrate the power of the notation The notation has been extended...

geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with...

are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and...

In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold...

Metric tensor Riemannian manifold Levi-Civita connection Curvature Riemann curvature tensor List of differential geometry topics Glossary of Riemannian and...

topogravitic tensor, which represents the spatial sectional curvatures, agrees with the electrogravitic tensor.) Looking back at our graph of the metric tensor, one...

u\rangle \langle v,v\rangle -\langle u,v\rangle ^{2}}} Here R is the Riemann curvature tensor, defined here by the convention R ( u , v ) w = ∇ u ∇ v w − ∇ v...

\nu }R_{\mu \nu }} is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as R μ ν = R α μ α ν . {\displaystyle...

electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a...

derivatives are often used in introductory derivations of the Riemann curvature tensor. Consider a curved rectangle with an infinitesimal vector δ {\displaystyle...

general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors — which represent curvature, hence the name...

two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense...

mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear...

called the Riemannian metric and the Riemann curvature tensor. For the surface (two-dimensional) case, the curvature at each point can be reduced to a number...