Are there vector fields which are gradients with respect to one metric but not another? 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 $M$, $$g(\text{grad}_g\,  f, Y) = df(Y).$$ 
 A: First note that a vector field $v$ can be written as a gradient of a function wrt. to a metric $g$ iff the corresponding one-form $\flat_g v$ is exact, i.e. $\flat_g v=df$ for some smooth function $f$, where $\flat_g$ is the isomorphism $TM\to T^{\ast}M$ induced by the metric $g$. If $M$ is contractible, any differential form $\omega$ is exact iff it is closed, i.e. $d\omega=0$. 
Thus, take any vector field $v$ on, say, $\mathbb{R}^2$ such that $d(\flat_g v)=0$. It is easy to perturb $g$ into another metric $h$ such as to achieve $d(\flat_h v)\neq 0$. 
A: 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 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.
