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 provide a gradient-flow characterisation for dissipative quantum systems. Namely, we show that finite-dimensional ergodic Lindblad equations admit a gradient flow structure for the von Neumann relative entropy if and only if the condition of bkm-detailed balance holds.
