I apologize if this question is too open for this forum.
Denote $g$ a positive definite matrix field (defined over, say, $\mathbb R^n$), and $g_{ij}$ its components in some basis (the same for all $g(x), x \in \mathbb R^n$). So $M$ is a smooth function that, to $x\in \mathbb R^n$, associates a positive definite matrix $g(x)$.
I am carrying out some computations, and they turn out to be valid only if:
$\sum_{t=1}^n \partial_t g_{jk} g_{ti} - \partial_t g_{ji}g_{tk} = 0$
where $\partial_t$ denotes differentiation wrt to the $t$-th variable (components in the same basis).
At first glance, this seems rather a high expectation. But I also know I am very ignorant of Riemann geometry and this could be some well-known invariant of a certain class of metric tensors. Understanding this constraint better could help me know if it is realistic to expect such an $g$ be produced in my circumstances, and perhaps how. None of my searches have provided any clues (similar expressions).