Fixing some notations about tensor calculus

Question

I'm studing some articles about Ricci Flow and Ricci Solitons. I realize the first thing to do is choose a notation, thus I'm using the Huai Dong Cao's notation (which is the same Ricgard Hamilton's notation), you can check here the computations

For example, the (4,0) version of the curvature tensor is given in local coordinates by $R_{ijkl}=g_{kh}R^h_{ijl}$, thus I lower the index to the third position.

If someone use the same notation or knows how to do the computations in this notation, I'd like to know how compute the divergence of a (4,0)-tensor: (Which is the correct)

$$g^{sl}\nabla_sT_{ijkl}$$ or $$g^{si}\nabla_sT_{ijkl}$$ or $$g^{sk}\nabla_sT_{ijkl}?$$

The last case is because I think in the Riemannian tensor.

Thanks!

Answer 1

All of those expressions represent different divergences of the 4-th order tensor (as well as the one you omitted, $g^{sj}\nabla_sT_{ijkl}$). Starting at 2nd-order tensors, you have to specify which index you are taking the divergence over. The symmetries of the tensor may reduce the number of distinct divergences, for example, for the symmetric stress-energy tensor the two divergences are equal: $\nabla_iT^{ij} = \nabla_iT^{ji}$.