Third rank curvature tensor?

1. Nov 10, 2014

copernicus1

Hi, I am curious about expressing the curvature of a manifold using a third-rank tensor. The fourth order Riemann tensor can be contracted to give the second order Ricci tensor and zeroth order Ricci scalar, but is there no way of obtaining a third- or first-order tensors, or would this simply not make sense for some reason?

Thanks!

2. Nov 16, 2014