Assume that $M$ is a differentiable manifold. Is there a Riemannian metric on $M$ such that the space of all gradient vector fields on $M$ would be closed under the Lie bracket?



The Lie algebra of gradient vector fields
If $M$ is the circle $S^1$ there is not such metric. Indeed, any Riemannian metric on $S^1$ is isometric to the one induced on a circle of radious $r$ of the plane centered at some point. Then take the angular "coordinate" $\theta$ and the functions $\cos(\theta)$ and $\sin(\theta)$ and check that the Lie bracket of their gradients is not a gradient. 


