Characterisation of gradient flows for a given functional

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 gα β g α β on M such that Y_β = gα β X^α Y β = g α β X α ? 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.