Metric Induced Curvature
The Riemann curvature tensor (or RiemannChristoffel tensor) is defined in terms of the connection by the relation,
where is the metric connection, given as
Thus, depends on the metric and its first and second derivatives.
At any point P of a manifold is a symmetric matrix of real numbers. Therefore, by standard matrix theory, there exists a transformation which reduces the matrix to diagonal form with every diagonal term either +1 or 1.
The excess of plus signs over minus signs in this form is called the signature of the metric. Assuming that the manifold is continuous and nonsingular, the signature is an invariant. In general, it will not be possible to find a coordinate system in which the metric reduces to this diagonal form everywhere. If, however, there does exist a coordinate system in which the metric reduces to diagonal form with 1 diagonal element everywhere, then the metric is called flat.
How does metric flatness relate to affine flatness in the case we are interested in, that is, when the connection is the metric connection? The answer is contained in the following result.
Theorem: A necessary and sufficient condition for a metric to be flat is that its Riemann tensor vanishes.
Necessary Condition Discussion:
Necessity follows from the fact that there exists a coordinate system in which the metric is diagonal with 1 diagonal element. Since the metric is constant everywhere, its partial derivatives vanish and therefore the metric connection vanishes as a consequence of the definition (7.2). Since vanishes everywhere, then so must its derivatives. The Riemann tensor therefore vanishes by the definition (7.1).
Sufficient Condition Discussion:
Since we are using the metric connection, we know that
(7.3)
This can be expanded to give the relation
(7.4)
from which we get


If the Riemann tensor vanishes, then by the Riemann curvature theorem concerning affine connections that was discussed in section 5, we know that there exists a special coordinate system in which the connection vanishes everywhere. From equation (7.5) it follows that
(7.6) 
. 

This means that the metric must be constant everywhere. Hence, it can be transformed into diagonal form with diagonal elements 1. Note that the result (7.5) expresses the ordinary derivatives of the metric in terms of the connection. This equation will prove useful.
Combining this metricinduced Riemann curvature theorem with the Riemann curvature theorem concerning affine connections, we see that if we use the metric connection then metric flatness coincides with affine flatness.
It follows immediately from the definition of the Riemann tensor
(7.7)
that it is antisymmetric on its last pair of indices :


The fact that the connection is symmetric leads to the identity
(7.9) 


Lowering the first index with the metric, then it is easy to establish, for example by using geodesic coordinates, that the lowered tensor is symmetric under interchange of the first and last pair of indices, that is,
(7.10) 


Combining this with equation (7.8), we see that the lowered tensor is antisymmetric on its first pair of indices as well:
(7.11)
Collecting these symmetries together, we see that the lowered curvature tensor satisfies
(7.12) 


(7.13) 


These symmetries considerably reduce the number of independent components; in fact, in ndimensions, the number is reduced as follows:
(7.14) 
. 

In addition to the algebraic identities, it can be shown, again most easily by using geodesic coordinates, that the curvature tensor satisfies a set of differential identities called the Bianchi identities:
Ricci Tensor
We can use the curvature tensor to define several other important tensors. The Ricci tensor is defined by the contraction


Since
(7.17) 


then we see that the Ricci tensor is symmetric.
(7.18) 


which by (7.16) is symmetric.
Ricci Scalar
A final contraction defines the Ricci scalar R by
(7.19) 


Einstein Tensor
These two tensors can be used to define the Einstein tensor
(7.20) 


which is also symmetric.
Contracted Bianchi Identities
By (7.15), the Einstein tensor can be shown to satisfy the contracted Bianchi identities
(7.21) 


Note that in different General Relativity textbooks authors will adopt different sign conventions for how the curvature tensor, and its associated contracted forms depend on the affine connections. In such books the Riemann tensor or the Ricci tensor can have the opposite signs to the definitions given above.
Notation: The book Schaum's Outline  Tensor Calculus by David Kay uses an unusual definition for the partial derivative of the metric. Kay uses the following definition.
and then also uses the uncommon definition
.
10/10/2002 12:20 PM