Suppose I have a smooth flow $\varphi(t)$ on some Riemannian manifold $(M,g)$ and I know that $\dot{\varphi}(t) = \textrm{grad }F$ for some smooth function $F$. If I smoothly modify the metric $g$, it's obvious that $\varphi$ will not usually still be a gradient flow of $F$, but will it still be the gradient flow of some other function $\hat{F}$? If so, is there a good way to determine what $\hat{F}$ is?
It seems like this should be true as long as $F$ is sufficiently nice. It's also worth mentioning that, while I'd like to find global results, I really only care if $\varphi$ is a gradient flow in some local chart.
At the very least, maybe someone could point me toward a useful book/article, or just some better terminology to google.
It's not true even locally: In $\mathbb R^2$, $X = (x, y)$ is a vector fields corresponding to $F = \frac 12(x^2 + y^2)$ via the standard metric. However, if you use the deformed one:
$$g_t = \begin{bmatrix} 1 & 0 \\ 0 & 1+tx \end{bmatrix}$$
then $X$ cannot be written as $\nabla_t F$ for any $F$, as the one form
$$\hat X_t(Y) = g_t(X, Y)$$
is not a closed one form when $t\neq 0$.