I initially asked on math.stackoverflow but have since come to understand this forum may be more appropriate, as this is indeed a question that arose in writing a research article.

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:

$\sum_{s=1}^n g_{si}\partial_s g_{jk} - g_{sk}\partial_s g_{ji} = 0$

where $\partial_t$ denotes differentiation wrt to the $t$-th variable (components in the same basis).

I first thought this condition to be unreasonable. I now have the suspicion this may be some kind of metric compatibility condition, but I don't manage to make sense of the results I find on this topic. 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.

Another lead may be to look at this as a PDE. This resembles the divergence of some field, but I am unfortunately not an expert of this either. Perhaps this corresponds to a well-known PDE.

Per suggestion, for $n=2$. For $j=1$:

$g_{11} \partial_1 g_{11} - g_{21} \partial_2 g_{11} = 0$

$g_{11} \partial_1 g_{12} - g_{22} \partial_2 g_{11} = 0$

$g_{12} \partial_1 g_{12} - g_{22} \partial_2 g_{12} = 0$

For $j=2$:

$g_{11} \partial_1 g_{21} - g_{21} \partial_2 g_{21} = 0$

$g_{11} \partial_1 g_{22} - g_{22} \partial_2 g_{21} = 0$

$g_{12} \partial_1 g_{22} - g_{22} \partial_2 g_{22} = 0$