Symmetry of Riemann Tensor: Investigating Rabmv

[space-time]

We know how objects such as the metric tensor and the Cristoffel symbol have symmetry to them (which is why g₁₂ = g₂₁ or \Gamma¹₁₂ = \Gamma¹₂₁)

Well I was wondering if the Riemann tensor R^a_bmv had any such symmetry. Are there any two or more particular indices that I could interchange and still get the same answer? If so, that will save me so much time when deriving these Riemann tensors.

 

[Matterwave]

Of course there is, there are actually many symmetries of the Riemann. In fact there is a whole section about it on Wikipedia.

See here: http://en.wikipedia.org/wiki/Riemann_curvature_tensor#Symmetries_and_identities

For example, in 4 dimensions the Riemann has only 20 independent components. But a rank 4 tensor in 4-D could have a possible 256 components! Good thing it has all these symmetries or else that is way too many components to keep track of!