# Riemann Tensor

## Contents

#### Notation

$g_{ab}$ is the metric, $\nabla_a$ is the covariant derivative, and $\partial_a$ is the partial derivative with respect to $x^a$.

Square brackets surrounding indices denote antisymmetrization, and round brackets denote symmetrization. For example, $U_{[a} V_{b]} \equiv (U_a V_b - U_b V_a)/2$ and $U_{(a} V_{b)} \equiv (U_a V_b + U_b V_a)/2$.

#### Christoffel symbols

$\begin{array}{rcl} \Gamma^a{}_{bc} & \equiv & g^{ad}(\partial_b g_{cd} + \partial_c g_{bd} - \partial_d g_{bc})/2 \\ \Gamma_{abc} & \equiv & g_{ad}\Gamma^d{}_{bc} \\ & = & (\partial_b g_{ca} + \partial_c g_{ba} - \partial_a g_{bc})/2 \\ \partial_a g_{bc} & = & 2 \Gamma_{(bc)a} \end{array}$

#### Definition of Riemann tensor

The Riemann tensor or the Riemann-Christoffel curvature tensor is a four-index tensor describing the curvature of Riemannian manifolds.

It is often used to derive other relativistic tensors such as the Ricci curvature tensor or the curvature scalar.

$\begin{array}{rcl} 2 \nabla_{[a} \nabla_{b]} V^c & = & R^c{}_{dab} V^d \\ 2 \nabla_{[a} \nabla_{b]} W_c & = & R_{abc}{}^d W_d \end{array}$

#### Riemann, Ricci, curvature scalar, Einstein tensor, and Weyl tensor:

Ricci and curvature scalar are both contractions of the Riemann tensor, simplifying it.

The scalar curvature is the trace of the Ricci curvature.

$\begin{array}{rcl} R^a{}_{bcd} & = & \partial_c \Gamma^a_{bd} - \partial_d \Gamma^a_{bc} + \Gamma^e_{bd}\Gamma^a_{ec} - \Gamma^e_{bc}\Gamma^a_{ed} \\ R_{bd} & \equiv & R^a{}_{bad} = \partial_a \Gamma^a_{bd} - \partial_d \Gamma^a_{ba} + \Gamma^e_{bd}\Gamma^a_{ea} - \Gamma^e_{ba}\Gamma^a_{ed} \\ R & \equiv & g^{bd} R_{bd} \\ G_{ab} & \equiv & R_{ab} - \frac{1}{2} R g_{ab} \\ W_{abcd} & \equiv & R_{abcd} - g_{a[c}R_{d]b} + g_{b[c}R_{d]a} + \frac{1}{3} R g_{a[c} g_{d]b} \end{array}$

Weyl curvature tensor represents the traceless component of the Riemann curvature tensor.

(The definition of the Weyl tensor depends on the number of spacetime dimensions $n$. Here it is assumed that $n=4$.)

#### Symmetries

For Riemann, the three symmetries of the curvature tensor are:

$\begin{array}{rcl} R_{bacd} & = & -R_{abcd} \\ R_{abdc} & = & -R_{abcd} \\ R_{cdab} & = & R_{abcd} \\ R_{a[bcd]} & = & 0 \end{array}$

The last symmetry, discovered by Ricci is called the first Bianchi identity or algebraic Bianchi identity. Given any tensor which satisfies these symmetries, one can completely describe a Riemannian manifold with the indicated curvature tensor at any point. The Ricci and Einstein tensors are symmetric.

#### Bianchi identities

$\begin{array}{rcl} \nabla_{[a} R_{bc]de} & = & 0 \\ \nabla_a G^{ab} & = & 0 \end{array}$