The Riemann tensor (Schutz 1985) 
, also
known as 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 as the Ricci curvature
tensor and scalar curvature can be defined
in terms of 
.
For the Levi-Civita connection of the metric tensor, the coordinate components are
 |
(1)
|
where 
are Christoffel symbols of the second
kind and 
is a comma derivative (Schmutzer 1968, p. 108;
Weinberg 1972; Carroll 1997, p. 76). The fully covariant tensor is 
.
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 independent components 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 at a point 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),
with the dimension list starting at 
(Carroll 1997, p. 80). On the space of algebraic curvature
tensors with a fixed nondegenerate metric tensor 
, the generic number of functionally independent
scalar invariants of 
under orthogonal changes of basis 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)
|
For 
dimensions, the Ricci-Weyl decomposition is
 |
(8)
|
Here, 
is the Ricci curvature tensor, 
is the scalar curvature,
and 
is the Weyl tensor (Carroll 1997, p. 82).
