Riemann Curvature Tensor - Symmetries and Identities

Symmetries and Identities

The Riemann curvature tensor has the following symmetries:

The last identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Bianchi identity below. (Also, if there is nonzero torsion, the first Bianchi identity becomes a differential identity of the torsion tensor.) These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has independent components.

Yet another useful identity follows from these three:

On a Riemannian manifold one has the covariant derivative and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form:

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

Skew symmetry

Interchange symmetry

First Bianchi identity

This is often written

where the brackets denote the antisymmetric part on the indicated indices. This is equivalent to the previous version of the identity because the Riemann tensor is already skew on its last two indices.

Second Bianchi identity

The semi-colon denotes a covariant derivative. Equivalently,

again using the antisymmetry on the last two indices of R.