Talk:Cotton tensor

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Making the definition readable by non-experts

I have read part of Berger Gauduchon Mazet long time ago and remember a few things about connections and the curvature tensor, and conventions in this field, but not all. The expression given for the coefficients of the Cotton tensor appears ambiguous to me.

Does $\nabla _{j}$ refer to the connection (covariant derivative) associated to g (and w.r.t. the vector field $X=\partial _{j}=\partial /\partial x^{j}$ ) or is it just the partial derivative w.r.t. j?
Does $\nabla _{j}\mathrm {Ric} _{ij}$ mean $(\nabla _{j}\mathrm {Ric} )_{ij}$ or $\nabla _{j}(\mathrm {Ric} _{ij})$ ? This is related to the previous question.
Similarly the absence of parentheses in $\nabla _{j}Rg_{ik}$ makes it ambiguous, and as a non-expert, I cannot guess which parenthesing (bracketing) was meant: $(\nabla _{j}(Rg))_{ik}$ , $\nabla _{j}(Rg_{ik})$ , $(\nabla _{j}R)g_{ik}$ , maybe another one I would have missed?

Arnaud Chéritat (talk) 10:04, 24 June 2021 (UTC)[reply]