Characterisation of gradient flows for a given functional

Morris Brooks, Jan Maas

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 r…