Is it possible for a vector field on a smooth manifold $M$ to be a gradient with respect to a Riemannian metric $g$, but not a gradient with respect to a different Riemannian metric $h$?


For completeness: the gradient of a smooth function $f:M\to \mathbb{R}$ with respect to the metric $g$ is the unique smooth vector field $\text{grad}\_g\, f$ on $M$ such that for all smooth vector fields $Y$ on $M$, $$g(\text{grad}\_g\, f, Y) = df(Y).$$ 



Are there vector fields which are gradients with respect to one metric but not another?
Consider the vector field $$X=(y-10x)\partial\_x-x\partial\_y$$ it is not a gradient vector field with respect to the standard Riemannian metric of $\mathbb{R}^2$ but it is a gradient vector field with respect to the Riemannian metric $$g=5dx\otimes dx-dx\otimes dy -dy\otimes dx +5dy\otimes dy$$


For this metric we have $X=grad\_g f$ where $f(x,y)=-1/2(49x^2-10xy+y^2)$
The motivation for consideration of a big number as $10$ in $X$ is the following:


**Note:** If we choose the vector field $$X=(y-\epsilon x)\partial\_x-x\partial\_y$$ where $|\epsilon|<2$ then there is no an analytic Riemannian metric $g$ with $X=\text{grad}\_g$. Because the orbits tend to the singularity at the origin, spirally. This means that the singularity is a focus singularity. On the other hand the Riemannian version of the [Thom gradient conjecture](https://en.wikipedia.org/wiki/Gradient_conjecture) says that if an orbit of an analytic gradient vector field tends to an isolated singularity then it must approach to the singularity in a specific direction. Of course a focus singularity violate this condition.


**Shahshahani Gradient:**
As an interesting example, please search "Shahshahani Gradient". This gradient corresponds to a Riemannian metric on the phase space of certain vector field which is not a gradient system with respect to the standard metric but is a gradient vector field with respect to the Shahshahani Riemannian metric. 


See here for some explanation on [Shahshahani metric.](https://books.google.com/books?id=AdKnOpCsH3gC&pg=PA52&lpg=PA52&dq=Shahshahani++Gradient++gradient+vector++field&source=bl&ots=huTRLFy49f&sig=LKs-EYCuLjnlK5t9KtOO18R2_fA&hl=en&sa=X&ved=0ahUKEwjZkOu_8bbbAhW0yqYKHUltC-g4ChDoAQgrMAI#v=onepage&q=Shahshahani%20%20Gradient%20%20gradient%20vector%20%20field&f=false)


