The Riemann tensor (Schutz 1985) 
, also known the Riemann-Christoffel
curvature tensor (Weinberg 1972, p. 133; Arfken 1985, p. 123) or Riemann
curvature tensor (Misner et al. 1973, p. 218), is a four-index tensor
that is useful in general relativity. Other important general relativistic tensors
such that the Ricci curvature tensor and
scalar curvature can be defined in terms of 
.
The Riemann tensor is in some sense the only tensor that can be constructed from the metric tensor and its first and second derivatives,
 |
(1)
|
where 
are Christoffel symbols of the first
kind and 
is a comma derivative (Schmutzer 1968, p. 108;
Weinberg 1972). In one dimension, 
. In four dimensions, there are 256 components. Making
use of the symmetry relations,
 |
(2)
|
the number of independent components is reduced to 36. Using the condition
 |
(3)
|
the number of coordinates reduces to 21. Finally, using
 |
(4)
|
20 independent components are left (Misner et al. 1973, pp. 220-221; Arfken 1985, pp. 123-124).
In general, the number of independent components in 
dimensions is given by
 |
(5)
|
the "four-dimensional pyramidal numbers," the first few values of which are 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, ... (OEIS A002415).
The number of scalars which can be constructed from 
and 
is
 |
(6)
|
(Weinberg 1972). The first few values are then 0, 1, 3, 14, 40, 90, 175, 308, 504, 780, ... (OEIS A050297).
In terms of the Jacobi tensor 
,
 |
(7)
|
Let
 |
(8)
|
where the quantity inside the 
is a Christoffel
symbol of the second kind. Then
 |
(9)
|
Broken down into its simplest decomposition in 
dimensions,
 |
(10)
|
Here, 
is the Ricci curvature tensor, 
is the scalar curvature,
and 
is the Weyl tensor.
