A Tensor sometimes known as the Riemann-Christoffel Tensor. Let
![\begin{displaymath}
\tilde D_s\equiv {\partial\over\partial x^s} - \sum_l \left\{{\matrix{s \quad u\cr l\cr}}\right\},
\end{displaymath}](r_1530.gif) |
(1) |
where the quantity inside the
is a
Christoffel Symbol of the Second Kind. Then
![\begin{displaymath}
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\}.
\end{displaymath}](r_1532.gif) |
(2) |
Broken down into its simplest decomposition in
-D,
|
|
|
(3) |
Here,
is the Ricci Tensor,
is the Curvature Scalar, and
is the
Weyl Tensor. In terms of the Jacobi Tensor
,
![\begin{displaymath}
R^\mu{}_{\alpha\nu\beta} = {\textstyle{2\over 3}} (J^\mu_{\nu\alpha\beta}-J^\mu_{\beta\alpha\nu}).
\end{displaymath}](r_1538.gif) |
(4) |
The Riemann tensor is the only tensor that can be constructed from the Metric Tensor and its first and second
derivatives,
![\begin{displaymath}
R^\alpha{}_{\beta\gamma\delta} =\Gamma^\alpha_{\beta\delta,\...
..._{\mu\delta}-\Gamma^\alpha_{\beta\mu}c_{\gamma\delta}{}^{\mu},
\end{displaymath}](r_1539.gif) |
(5) |
where
are Connection Coefficients and
are Commutation
Coefficients. The number of independent coordinates in
-D is
![\begin{displaymath}
C_n\equiv {\textstyle{1\over 12}}n^2(n^2-1),
\end{displaymath}](r_1541.gif) |
(6) |
and the number of Scalars which can be constructed from
and
is
![\begin{displaymath}
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}
\end{displaymath}](r_1544.gif) |
(7) |
In 1-D,
.
![$n$](r_1.gif) |
![$C_n$](r_1546.gif) |
![$S_n$](r_1547.gif) |
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
© 1996-9 Eric W. Weisstein
1999-05-25