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Let $(M^2,g)$ be a closed surface and let $X,Y\in C^{\infty}(TM)$ such that $[X,Y] = 0$. I am working on a problem where I have to deal with the following term $$g(\nabla_XX,\nabla_YY) - |\nabla_XY|^2_g.$$

If one thinks about the connection matrix of $\nabla$ this looks a lot like something related to the determinant of this matrix. Is this expression related to some characteristic class? For example, in the case $T^2$ is a torus, should its integral be zero?

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    $\begingroup$ Do you know tensorial properties of this thing? Is it a tensor or differential operator? Of what order if the latter? I can try to calculate it by myself but it is the first thing to check - if it is a tensor it should be a full contraction of covariant derivatives of curvature. $\endgroup$ – Lev Soukhanov Feb 5 '20 at 19:38
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    $\begingroup$ I think invariant differential operators also should have some classification. You can look up this paper for the result im citing in the previous comment: F. Prüfer, F. Tricerri & L. Vanhecke, “Curvature invariants, differential operators and local homogeneity" $\endgroup$ – Lev Soukhanov Feb 5 '20 at 19:41

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