Riemann curvature tensor

Let (M, g) be a Riemannian or pseudo-Riemannian manifold, and ${\displaystyle {\mathfrak {X}}(M)}$ be the space of all vector fields on M. We define the Riemann curvature tensor as a map ${\displaystyle R\colon {\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\rightarrow {\mathfrak {X}}(M)}$ by the following formula[2] where ${\displaystyle \nabla }$ is the Levi-Civita connection:

${\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z}$

or equivalently

${\displaystyle R(X,Y)=[\nabla _{X},\nabla _{Y}]-\nabla _{[X,Y]}}$

where [X, Y] is the Lie bracket of vector fields and ${\displaystyle [\nabla _{X},\nabla _{Y}]}$ is a commutator of differential operators. It turns out that the right-hand side actually only depends on the value of the vector fields ${\displaystyle X,Y,Z}$ at a given point, which is notable since the covariant derivative of a vector field also depends on the field values in a neighborhood of the point. Hence, ${\displaystyle R}$ is a ${\displaystyle (3,1)}$-tensor field. For fixed ${\displaystyle X,Y}$, the linear transformation ${\displaystyle Z\mapsto R(X,Y)Z}$ is also called the curvature transformation or endomorphism. Occasionally, the curvature tensor is defined with the opposite sign.

The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space).

Since the Levi-Civita connection is torsion-free, the curvature can also be expressed in terms of the second covariant derivative[3]

${\textstyle \nabla _{X,Y}^{2}Z=\nabla _{X}\nabla _{Y}Z-\nabla _{\nabla _{X}Y}Z}$

as

${\displaystyle R(X,Y)=\nabla _{X,Y}^{2}-\nabla _{Y,X}^{2}}$

Thus, the curvature tensor measures the noncommutativity of the second covariant derivative. In abstract index notation,

$${\displaystyle R^{d}{}_{cab}Z^{c}=\nabla _{a}\nabla _{b}Z^{d}-\nabla _{b}\nabla _{a}Z^{d}.}$$

The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector ${\displaystyle A_{\nu }}$ with itself:[4][5]

${\displaystyle A_{\nu$ ;\rho \sigma }-A_{\nu ;\sigma \rho }=A_{\beta }R^{\beta }{}_{\nu \rho \sigma }.}

This formula is often called the Ricci identity.[6] This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor.[7] This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows [8]

${\displaystyle {\begin{aligned}&\nabla _{\delta }\nabla _{\gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}-\nabla _{\gamma }\nabla _{\delta }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}\\[3pt]={}&R^{\alpha _{1}}{}_{\rho \delta \gamma }T^{\rho \alpha _{2}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s}}+\ldots +R^{\alpha _{r}}{}_{\rho \delta \gamma }T^{\alpha _{1}\cdots \alpha _{r-1}\rho }{}_{\beta _{1}\cdots \beta _{s}}-R^{\sigma }{}_{\beta _{1}\delta \gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\sigma \beta _{2}\cdots \beta _{s}}-\ldots -R^{\sigma }{}_{\beta _{s}\delta \gamma }T^{\alpha _{1}\cdots \alpha _{r}}{}_{\beta _{1}\cdots \beta _{s-1}\sigma }\end{aligned}}}$

This formula also applies to tensor densities without alteration, because for the Levi-Civita (not generic) connection one gets:[6]

${\displaystyle \nabla _{\mu }\left({\sqrt {g}}\right)\equiv \left({\sqrt {g}}\right)_{;\mu }=0,}$

where

${\displaystyle g=\left|\det \left(g_{\mu \nu }\right)\right|.}$

It is sometimes convenient to also define the purely covariant version of the curvature tensor by

${\displaystyle R_{\rho \sigma \mu \nu }=g_{\rho \zeta }R^{\zeta }{}_{\sigma \mu \nu }.}$

Converting to the tensor index notation, the Riemann curvature tensor is given by

${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=dx^{\rho }\left(R\left(\partial _{\mu },\partial _{\nu }\right)\partial _{\sigma }\right)}$

where ${\displaystyle \partial _{\mu }=\partial /\partial x^{\mu }}$ are the coordinate vector fields. The above expression can be written using Christoffel symbols:

${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$

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

The Riemann curvature tensor has the following symmetries and identities:

Skew symmetry	${\displaystyle R(u,v)=-R(v,u)}$	${\displaystyle R_{abcd}=-R_{abdc}\Leftrightarrow R_{ab(cd)}=0}$
Skew symmetry	${\displaystyle \langle R(u,v)w,z\rangle =-\langle R(u,v)z,w\rangle }$	${\displaystyle R_{abcd}=-R_{bacd}\Leftrightarrow R_{(ab)cd}=0}$
First (algebraic) Bianchi identity	${\displaystyle R(u,v)w+R(v,w)u+R(w,u)v=0}$	${\displaystyle R_{abcd}+R_{acdb}+R_{adbc}=0\Leftrightarrow R_{a[bcd]}=0}$
Interchange symmetry	${\displaystyle \langle R(u,v)w,z\rangle =\langle R(w,z)u,v\rangle }$	${\displaystyle R_{abcd}=R_{cdab}}$
Second (differential) Bianchi identity	${\displaystyle \left(\nabla _{u}R\right)(v,w)+\left(\nabla _{v}R\right)(w,u)+\left(\nabla _{w}R\right)(u,v)=0}$	${\displaystyle R_{abcd;e}+R_{abde;c}+R_{abec;d}=0\Leftrightarrow R_{ab[cd;e]}=0}$

where the bracket ${\displaystyle \langle ,\rangle }$ refers to the inner product on the tangent space induced by the metric tensor and the brackets and parentheses on the indices denote the antisymmetrization and symmetrization operators, respectively. If there is nonzero torsion, the Bianchi identities involve the torsion tensor.

The first (algebraic) Bianchi identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the differential Bianchi identity.

The first 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 ${\displaystyle n^{2}\left(n^{2}-1\right)/12}$ independent components.[9] Interchange symmetry follows from these. The algebraic symmetries are also equivalent to saying that R belongs to the image of the Young symmetrizer corresponding to the partition 2+2.

On a Riemannian manifold one has the covariant derivative ${\displaystyle \nabla _{u}R}$ and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form of the last identity in the table.

The Ricci curvature tensor is the contraction of the first and third indices of the Riemann tensor.

${\displaystyle \underbrace {R_{ab}} _{\text{Ricci}}\equiv \underbrace {R^{c}{}_{acb}} _{\text{Riemann}}=g^{cd}\underbrace {R_{cadb}} _{\text{Riemann}}}$

• Introduction to the mathematics of general relativity
• Decomposition of the Riemann curvature tensor
• Curvature of Riemannian manifolds
• Ricci curvature tensor