In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics that measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
It is a central mathematical tool in the theory of general relativity, the modern theory of gravity. The curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.
Definition
Let <math>(M, g)</math> be a Riemannian or pseudo-Riemannian manifold, and <math>\mathfrak{X}(M)</math> be the space of all vector fields on <math>M</math>. We define the Riemann curvature tensor as a map <math>\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)</math> by the following formula where <math>\nabla</math> is the Levi-Civita connection:
: <math>R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z</math>
or equivalently
: <math>R(X, Y) = [\nabla_X,\nabla_Y] - \nabla_{[X, Y]} </math>
where <math>[X,Y]</math> is the Lie bracket of vector fields and <math>[\nabla_X,\nabla_Y] </math> is a commutator of differential operators. It turns out that the right-hand side actually only depends on the value of the vector fields <math>X, Y, Z</math> 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, <math>R</math> is a <math>(1,3)</math>-tensor field. For fixed <math>X,Y</math>, the linear transformation <math>Z \mapsto R(X, Y)Z</math> 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, its curvature can also be expressed in terms of the second covariant derivative
: <math display="inline">\nabla^2_{X,Y} Z = \nabla_X\nabla_Y Z - \nabla_{\nabla_X Y}Z </math>
which depends only on the values of <math>X, Y</math> at a point.
The curvature can then be written as
: <math>R(X, Y) = \nabla^2_{X,Y} - \nabla^2_{Y,X} </math>
Thus, the curvature tensor measures the noncommutativity of the second covariant derivative. In abstract index notation, <math display="block">R^d{}_{cab} Z^c = \nabla_a \nabla_b Z^d - \nabla_b \nabla_a Z^d . </math>The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector <math>A_{\nu}</math> with itself:
: <math>A_{\nu;\rho\sigma} - A_{\nu;\sigma\rho} = A_{\beta} R^{\beta}{}_{\nu\rho\sigma}.</math>
This formula is often called the Ricci identity. This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor. This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows
: <math>\begin{align}
&\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{align}</math>
This formula also applies to tensor densities without alteration, because for the Levi-Civita (not generic) connection one gets: 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 <math> \nabla_u R </math> and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form of the last identity in the table.
Ricci curvature
The Ricci curvature tensor is the contraction of the first and third indices of the Riemann tensor.
: <math>
\underbrace{R_{ab_{\text{Ricci
\equiv R^c{}_{acb}
= g^{cd} \underbrace{R_{cadb_{\text{Riemann
</math>
Special cases
Surfaces
For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor has only one independent component, which means that the Ricci scalar completely determines the Riemann tensor. There is only one valid expression for the Riemann tensor which fits the required symmetries:
: <math>R_{abcd} = f(R) \left(g_{ac}g_{db} - g_{ad}g_{cb}\right)</math>
and by contracting with the metric twice we find the explicit form:
: <math>R_{abcd} = K\left(g_{ac}g_{db} - g_{ad}g_{cb}\right) ,</math>
where <math>g_{ab}</math> is the metric tensor and <math>K = R/2</math> is a function called the Gaussian curvature and <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math> take values either 1 or 2. The Riemann tensor has only one functionally 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 surface is simply given by
: <math>R_{ab} = Kg_{ab}.</math>
Space forms
A Riemannian manifold is a space form if its sectional curvature is equal to a constant <math>K</math>. The Riemann tensor of a space form is given by
: <math>R_{abcd} = K\left(g_{ac}g_{db} - g_{ad}g_{cb}\right).</math>
Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function <math>K</math>, then the Bianchi identities imply that <math>K</math> is constant and thus that the manifold is (locally) a space form.
See also
- Introduction to the mathematics of general relativity
- Decomposition of the Riemann curvature tensor
- Curvature of Riemannian manifolds
- Ricci curvature tensor