General relativity
${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational redshift Gravitational time dilation Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Formalisms ADM BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Einstein–Rosen waves Wormhole Gödel Kerr Kerr–Newman Kerr–Newman–de Sitter Kasner Lemaître–Tolman Taub–NUT Milne Robertson–Walker Oppenheimer–Snyder pp-wave van Stockum dust Weyl−Lewis−Papapetrou Hartle–Thorne
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedmann Milne Zwicky Lemaître Oppenheimer Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Raychaudhuri Taylor Hulse van Stockum Taub Newman Yau Thorne others
Physics portal  Category
v t e

In themathematical field ofdifferential geometry, theRiemann curvature tensor orRiemann–Christoffel tensor (afterBernhard Riemann andElwin Bruno Christoffel) is the most common way used to express thecurvature of Riemannian manifolds. It assigns atensor to each point of aRiemannian manifold (i.e., it is atensor field). It is a local invariant of Riemannian metrics that measures the failure of the secondcovariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it isflat, i.e. locallyisometric to theEuclidean space.^[1] The curvature tensor can also be defined for anypseudo-Riemannian manifold, or indeed any manifold equipped with anaffine connection.

It is a central mathematical tool in the theory ofgeneral relativity, the modern theory ofgravity. The curvature ofspacetime is in principle observable via thegeodesic deviation equation. The curvature tensor represents thetidal force experienced by a rigid body moving along ageodesic in a sense made precise by theJacobi equation.

Let${\displaystyle (M,g)}$ be aRiemannian orpseudo-Riemannian manifold, and${\displaystyle \mathfrak{X}(M)}$ be the space of allvector fields on${\displaystyle M}$. We define theRiemann curvature tensor as a map${\displaystyle \mathfrak{X}(M)\times \mathfrak{X}(M)\times \mathfrak{X}(M)\to \mathfrak{X}(M)}$ by the following formula^[2] where${\displaystyle \mathrm{\nabla }}$ is theLevi-Civita connection:

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

or equivalently

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

where${\displaystyle [X,Y]}$ is theLie bracket of vector fields and${\displaystyle [{\mathrm{\nabla }}_X,{\mathrm{\nabla }}_Y]}$ 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${\displaystyle (1,3)}$-tensor field. For fixed${\displaystyle X,Y}$, the linear transformation${\displaystyle Z\mapsto R(X,Y)Z}$ is also called thecurvature transformation orendomorphism. Occasionally, the curvature tensor is defined with the opposite sign.

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

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

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

which depends only on the values of${\displaystyle X,Y}$ at a point.The curvature can then be written as

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

Thus, the curvature tensor measures the noncommutativity of the second covariant derivative. Inabstract index notation,$${\displaystyle R^d{}_{cab}{Z}^c={\mathrm{\nabla }}_a{\mathrm{\nabla }}_b{Z}^d-{\mathrm{\nabla }}_b{\mathrm{\nabla }}_a{Z}^d.}$$The Riemann curvature tensor is also thecommutator 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 theRicci identity.^[6] This is the classical method used byRicci andLevi-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]

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

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

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

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

One can see the effects of curved space by comparing a tennis court and the Earth. Start at the lower right corner of the tennis court, with a racket held out towards north. Then while walking around the outline of the court, at each step make sure the tennis racket is maintained in the same orientation, parallel to its previous positions. Once the loop is complete the tennis racket will be parallel to its initial starting position. This is because tennis courts are built so the surface is flat. On the other hand, the surface of the Earth is curved: we can complete a loop on the surface of the Earth. Starting at the equator, point a tennis racket north along the surface of the Earth. Once again the tennis racket should always remain parallel to its previous position, using the local plane of the horizon as a reference. For this path, first walk to the north pole, then walk sideways (i.e. without turning), then down to the equator, and finally walk backwards to your starting position. Now the tennis racket will be pointing towards the west, even though when you began your journey it pointed north and you never turned your body. This process is akin toparallel transporting a vector along the path and the difference identifies how lines which appear "straight" are only "straight" locally. Each time a loop is completed the tennis racket will be deflected further from its initial position by an amount depending on the distance and the curvature of the surface. It is possible to identify paths along a curved surface where parallel transport works as it does on flat space. These are thegeodesics of the space, for example any segment of a great circle of a sphere.

The concept of a curved space in mathematics differs from conversational usage. For example, if the above process was completed on a cylinder one would find that it is not curved overall as the curvature around the cylinder cancels with the flatness along the cylinder, which is a consequence ofGaussian curvature and Gauss'sTheorema Egregium. A familiar example of this is a floppy pizza slice, which will remain rigid along its length if it is curved along its width.

The Riemann curvature tensor is a way to capture a measure of the intrinsic curvature. When you write it down in terms of its components (like writing down the components of a vector), it consists of a multi-dimensional array of sums and products of partial derivatives (some of those partial derivatives can be thought of as akin to capturing the curvature imposed upon someone walking in straight lines on a curved surface).

When a vector in a Euclidean space isparallel transported around a loop, it will again point in the initial direction after returning to its original position. However, this property does not hold in the general case. The Riemann curvature tensor directly measures the failure of this in a generalRiemannian manifold. This failure is known as the non-holonomy of the manifold.

Let${\displaystyle x_t}$ be a curve in a Riemannian manifold${\displaystyle M}$. Denote by${\displaystyle {\tau }_{x_t}:T_{x_0}M\to T_{x_t}M}$ the parallel transport map along${\displaystyle x_t}$. The parallel transport maps are related to thecovariant derivative by

${\displaystyle {\mathrm{\nabla }}_{{\dot{x}}_0}Y=\underset{h\to 0}{lim}\frac{1}{h}\left({\tau }_{x_h}^{-1}\left(Y_{x_h}\right)-Y_{x_0}\right)={\frac{d}{dt}\left({\tau }_{x_t}^{-1}(Y_{x_t})\right)|}_{t=0}}$

for eachvector field${\displaystyle Y}$ defined along the curve.

Suppose that${\displaystyle X}$ and${\displaystyle Y}$ are a pair of commuting vector fields. Each of these fields generates a one-parameter group of diffeomorphisms in a neighborhood of${\displaystyle x_0}$. Denote by${\displaystyle {\tau }_{tX}}$ and${\displaystyle {\tau }_{tY}}$, respectively, the parallel transports along the flows of${\displaystyle X}$ and${\displaystyle Y}$ for time${\displaystyle t}$. Parallel transport of a vector${\displaystyle Z\in T_{x_0}M}$ around the quadrilateral with sides${\displaystyle tY}$,${\displaystyle sX}$,${\displaystyle -tY}$,${\displaystyle -sX}$ is given by

${\displaystyle {\tau }_{sX}^{-1}{\tau }_{tY}^{-1}{\tau }_{sX}{\tau }_{tY}Z.}$

The difference between this and${\displaystyle Z}$ measures the failure of parallel transport to return${\displaystyle Z}$ to its original position in the tangent space${\displaystyle T_{x_0}M}$. Shrinking the loop by sending${\displaystyle s,t\to 0}$ gives the infinitesimal description of this deviation:

${\displaystyle {\frac{d}{ds}\frac{d}{dt}{\tau }_{sX}^{-1}{\tau }_{tY}^{-1}{\tau }_{sX}{\tau }_{tY}Z|}_{s=t=0}=\left({\mathrm{\nabla }}_X{\mathrm{\nabla }}_Y-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_X-{\mathrm{\nabla }}_{[X,Y]}\right)Z=R(X,Y)Z}$

where${\displaystyle R}$ is the Riemann curvature tensor.

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

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

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

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

(See alsoList of formulas in Riemannian geometry).

The Riemann curvature tensor has the following symmetries and identities:

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

where the bracket${\displaystyle ⟨,⟩}$ refers to the inner product on the tangent space induced by themetric tensor andthe brackets and parentheses on the indices denote theantisymmetrization andsymmetrization operators, respectively. If there is nonzerotorsion, the Bianchi identities involve thetorsion tensor.

The first (algebraic) Bianchi identity was discovered byRicci, but is often called thefirst Bianchi identity oralgebraic Bianchi identity, because it looks similar to the differentialBianchi identity.^{[citation needed]}

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 thatR belongs to the image of theYoung symmetrizer corresponding to the partition 2+2.

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

TheRicci curvature tensor is thecontraction of the first and third indices of the Riemann tensor.

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

For a two-dimensionalsurface, the Bianchi identities imply that the Riemann tensor has only one independent component, which means that theRicci scalar completely determines the Riemann tensor. There is only one valid expression for the Riemann tensor which fits the required symmetries:

${\displaystyle R_{abcd}=f(R)\left(g_{ac}{g}_{db}-g_{ad}{g}_{cb}\right)}$

and by contracting with the metric twice we find the explicit form:

${\displaystyle R_{abcd}=K\left(g_{ac}{g}_{db}-g_{ad}{g}_{cb}\right),}$

where${\displaystyle g_{ab}}$ is themetric tensor and${\displaystyle K=R/2}$ is a function called theGaussian curvature and${\displaystyle a}$,${\displaystyle b}$,${\displaystyle c}$ and${\displaystyle d}$ take values either 1 or 2. The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides with thesectional curvature of the surface. It is also exactly half thescalar curvature of the 2-manifold, while theRicci curvature tensor of the surface is simply given by

${\displaystyle R_{ab}=K{g}_{ab}.}$

A Riemannian manifold is aspace form if itssectional curvature is equal to a constant${\displaystyle K}$. The Riemann tensor of a space form is given by

${\displaystyle R_{abcd}=K\left(g_{ac}{g}_{db}-g_{ad}{g}_{cb}\right).}$

Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function${\displaystyle K}$, then the Bianchi identities imply that${\displaystyle K}$ is constant and thus that the manifold is (locally) a space form.

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

• Theorems about circles
• Bernhard Riemann
• Curvature (mathematics)
• Differential geometry
• Riemannian geometry
• Riemannian manifolds
• Tensors in general relativity

• CS1 errors: ISBN date
• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from March 2022