A Tensor sometimes known as the RiemannChristoffel 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 1D, .



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
© 19969 Eric W. Weisstein
19990525