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Let $(M,g)$ be a riemannian manifold with Levi-Civita connection $\nabla$. When does it exist locally a potential function $\phi$ for the metric $g$? When is $g$, the Hessian of $\phi$?

$$g(X,Y)= (XY+YX-\nabla_X Y- \nabla_Y X)\phi$$

From your definition of potential we have

$$g(X,Y)=-\big((\nabla_XY)+(\nabla_YX)\big)\phi$$

The formula has to apply (locally) for any vectors $X$ and $Y$. Consider a time-like geodesic with tangent vector $U$. Choose $X=U$, $Y=U$. We have a contradiction.

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