Name of the tensor

Question

Please anyone tell me if the given tensor has any specific name in literature $$g_{jk}\nabla_iR-g_{ik}\nabla_jR,$$ where $R$ is the scalar curvature.

Answer 1

For a tensor (or any other quantity) to have a specific name, there must be a strong reason for that. For example, such a tensor could play a significant role in some theory, so that people need to refer to it quite often, and saying something like "the anti-symmetrization with respect to the two last indices of the tensor product of the Riemannian metric and the covariant derivative of the scalar curvature" (which is what your tensor is) would be too tiresome.

You can compare your tensor with the Cotton tensor, which indeed plays an important role in the Riemannian geometry.

The formula in your question is an example of a Weyl metric invariant, that is a linear combination of (compatible) partial contractions of tensor products of some number of the metric tensor and some number of (multiple) covariant derivatives of the Riemann tensor. It can be written off in a coordinate patch as a polynomial expression involving (multiple) partial derivatives of the components of the metric tensor and some other technical objects. If such an expression exhibits certain naturality properties, it is called a metric invariant. There is a nontrivial theorem (due to Hermann Weyl), stating that all such metric invariants can be obtained as Weyl metric invariants. If you want to find more precise statements, please see my answer here, and references therein.