A Tensor sometimes known as the Riemann-Christoffel Tensor. Let
|
(1) |
where the quantity inside the
is a
Christoffel Symbol of the Second Kind. Then
|
(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
,
|
(4) |
The Riemann tensor is the only tensor that can be constructed from the Metric Tensor and its first and second
derivatives,
|
(5) |
where are Connection Coefficients and are Commutation
Coefficients. The number of independent coordinates in -D is
|
(6) |
and the number of Scalars which can be constructed from
and is
|
(7) |
In 1-D, .
|
|
|
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