Let $X$ be a vector field and $Y$ be a co-vector field on a smooth manifold
$M$. Does there exist a smooth Riemannian metric $g_{\alpha \beta}$ on $M$ such
that $Y_\beta = g_{\alpha \beta} X^\alpha$? The main result of this note gives
necessary and sufficient conditions for this to be true. As an application of
this result we show that a finite-dimensional ergodic Lindblad equation admits
a gradient flow structure for the von Neumann relative entropy if and only if
the condition of BKM-detailed balance holds.

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