Riemann Tensor

A Tensor sometimes known as the Riemann-Christoffel Tensor. Let

$$\tilde D_s\equiv {\partial\over\partial x^s} - \sum_l \left\{{\matrix{s \quad u\cr l\cr}}\right\},$$ (1)

where the quantity inside the $\left\{{\matrix{s\quad u\cr l\cr}}\right\}$ is a Christoffel Symbol of the Second Kind. Then

$$R_{pqrs} \equiv \tilde D_q\left\{{\matrix{p\quad r\cr s\cr}}\right\} - \tilde D_r\left\{{\matrix{r\quad q\cr s\cr}}\right\}.$$ (2)

Broken down into its simplest decomposition in $N$-D,

$R_{\lambda\mu\nu\kappa}={1\over N-2} (g_{\lambda\nu}R_{\mu\kappa}-g_{\lambda\kappa} R_{\mu\nu}-g_{\mu\nu}R_{\lambda\kappa}+g_{\mu\kappa}R_{\lambda\nu})$
$- {R\over (N-1)(N-2)} (g_{\lambda\nu}g_{\mu\kappa}-g_{\lambda\kappa}g_{\mu\nu})+ C_{\lambda\mu\nu\kappa}.\quad$	(3)

Here, $R_{\mu\nu}$ is the Ricci Tensor, $R$ is the Curvature Scalar, and $C_{\lambda\mu\nu\kappa}$ is the Weyl Tensor. In terms of the Jacobi Tensor ${J^\mu}_{\nu\alpha\beta}$,

$$R^\mu{}_{\alpha\nu\beta} = {\textstyle{2\over 3}} (J^\mu_{\nu\alpha\beta}-J^\mu_{\beta\alpha\nu}).$$ (4)

The Riemann tensor is the only tensor that can be constructed from the Metric Tensor and its first and second derivatives,

$$R^\alpha{}_{\beta\gamma\delta} =\Gamma^\alpha_{\beta\delta,\... ..._{\mu\delta}-\Gamma^\alpha_{\beta\mu}c_{\gamma\delta}{}^{\mu},$$ (5)

where $\Gamma$ are Connection Coefficients and $c$ are Commutation Coefficients. The number of independent coordinates in $n$-D is

$$C_n\equiv {\textstyle{1\over 12}}n^2(n^2-1),$$ (6)

and the number of Scalars which can be constructed from $R_{\lambda\mu\nu\kappa}$ and $g_{\mu\nu}$ is

$$S_n\equiv \cases{ 1 & for $n=2$\cr {\textstyle{1\over 12}} n(n-1)(n-2)(n+3) & for $n=1, n>2$.\cr}$$ (7)

In 1-D, $R_{1111}=0$.

$n$	$C_n$	$S_n$
1	0	0
2	1	1
3	6	3
4	20	14

See also Bianchi Identities, Christoffel Symbol of the Second Kind, Commutation Coefficient, Connection Coefficient, Curvature Scalar, Gaussian Curvature, Jacobi Tensor, Petrov Notation, Ricci Tensor, Weyl Tensor