Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express curvature of Riemannian manifolds. It is one of many things named after Bernhard Riemann and Elwin Bruno Christoffel. The curvature tensor is given in terms of a Levi-Civita connection by the following formula:

:R(u,v)w= abla_u abla_v w - abla_v abla_u w - abla_{ [u,v] } w .

Here R(u,v) is a linear transformation of the tangent space of the manifold; it is linear in each argument.

NB. Some authors define the curvature tensor with the opposite sign.

If u=partial/partial x^i and v=partial/partial x^j are coordinate vector fields then [u,v] =0 and therefore the formula simplifies to :R(u,v)w= abla_u abla_v w - abla_v abla_u w i.e. the curvature tensor measures "noncommutativity of the covariant derivative".

The linear transformation wmapsto R(u,v)w is also called the curvature transformation or endomorphism.

The Riemann curvature tensor, especially in its coordinate expression (see below), is a central mathematical tool of general relativity, the modern theory of gravity.

Coordinate expression

In local coordinates x^mu the Riemann curvature tensor is given by:{R^ ho}_{sigmamu u} = dx^ ho(R(partial_{mu},partial_{ u})partial_{sigma})where partial_{mu} = partial/partial x^{mu} are the coordinate vector fields. The above expression can be written using Christoffel symbols::{R^ ho}_{sigmamu u} = partial_muGamma^ ho_{ usigma} - partial_ uGamma^ ho_{musigma} + Gamma^ ho_{mulambda}Gamma^lambda_{ usigma} - Gamma^ ho_{ ulambda}Gamma^lambda_{musigma}

(see also the list of formulas in Riemannian geometry).

The transformation of a vector V^mu after circling an infinitesimal rectangle dx^ u dx^sigma is:delta V^mu = R^mu_{ usigma au} dx^ u dx^sigma V^ au.

Also define the purely covariant version by:R_{ hosigmamu u} = g_{ ho zeta} {R^zeta}_{sigmamu u} .

ymmetries and identities

The Riemann curvature tensor has the following symmetries::R(u,v)=-R(v,u)^{}_{}:langle R(u,v)w,z angle=-langle R(u,v)z,w angle^{}_{}:R(u,v)w+R(v,w)u+R(w,u)v=0 ^{}_{}.

The last identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Bianchi identity below. These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has n^2(n^2-1)/12 independent components.

Yet another useful identity follows from these three::langle R(u,v)w,z angle=langle R(w,z)u,v angle^{}_{}.

The Bianchi identity (often called the second Bianchi identity or differential Bianchi identity)involves the covariant derivative:

: abla_uR(v,w)+ abla_vR(w,u)+ abla_w R(u,v) = 0.

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

:R_{abcd}^{}=-R_{bacd}=-R_{abdc}

:R_{abcd}^{}=R_{cdab}

:R_{a [bcd] }^{}=0 (first Bianchi identity)

:R_{ab [cd;e] }^{}=0 (second Bianchi identity)

where the square brackets denote cyclic symmetrisation over the indices and the semi-colon is a covariant derivative.

For surfaces

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as:R_{abcd}^{}=K(g_{ac}g_{db}- g_{ad}g_{cb} ) where g_{ab} is the metric tensor and K is a function called the Gaussian curvature and "a", "b", "c" and "d" take values either "1" or "2". As expected we see that the Riemann curvature tensor only has one independent component.

The Gaussian curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surfaceis simply given by :operatorname{Ric}_{ab} = Kg_{ab}.

ee also

*Curvature of Riemannian manifolds
*Sectional curvature
*Curvature form
*Basic introduction to the mathematics of curved spacetime
*Holonomy


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Curvature tensor — In differential geometry, the term curvature tensor may refer to: the Riemann curvature tensor of a Riemannian manifold see also Curvature of Riemannian manifolds; the curvature of an affine connection or covariant derivative (on tensors); the… …   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

  • Tensor — For other uses, see Tensor (disambiguation). Note that in common usage, the term tensor is also used to refer to a tensor field. Stress, a second order tensor. The tensor s components, in a three dimensional Cartesian coordinate system, form the… …   Wikipedia

  • Curvature — In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this …   Wikipedia

  • Curvature form — In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry. Contents 1 Definition 1.1 Curvature… …   Wikipedia

  • Tensor-Diagramm-Notation — Die penrosesche graphische Notation – auch als penrosesche diagrammatische Notation, Tensor Diagramm Notation oder auch einfach Penrose Notation bezeichnet – ist eine von Roger Penrose vorgeschlagene Notation in der Physik und Mathematik, um eine …   Deutsch Wikipedia

  • Tensor contraction — In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components… …   Wikipedia

  • Riemann tensor (general relativity) — The Riemann tensor (general relativity) is a mathematical object that describes gravitation and its effects in Einstein s theory of general relativity. Curvature and geodesic deviationThe Riemann tensor can be used to express the idea of… …   Wikipedia

  • Tensor de curvatura — En geometría diferencial, el tensor de curvatura de Riemann , o simplemente tensor de curvatura o tensor de Riemann, supone una generalización del concepto de curvatura de Gauss, definido para superficies, a variedades de dimensiones arbitrarias …   Wikipedia Español

  • Curvature invariant (general relativity) — Curvature invariants in general relativity are a set of scalars called curvature invariants that arise in general relativity. They are formed from the Riemann, Weyl and Ricci tensors which represent curvature and possibly operations on them such… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.