# Riemann curvature tensor

In the

mathematical field ofdifferential geometry , the**Riemann curvature tensor**or**Riemann–Christoffel tensor**is the most standard way to expresscurvature of Riemannian manifolds . It is one of many things named afterBernhard Riemann andElwin Bruno Christoffel . The curvature tensor is given in terms of aLevi-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 ofgravity .**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 themetric tensor and $K$ is a function called theGaussian 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 thescalar curvature of the 2-manifold, while theRicci 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.*

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