Let $\phi(x)$ be a scalar field and $\nabla_{a}$ denote the Levi-Civita covariant derivative operator of a metric tensor $g_{a b}$. Show that

$$\nabla_{a} \nabla_{b} \phi=\nabla_{b} \nabla_{a} \phi$$

If the Ricci tensor, $R_{a b}$, of the metric $g_{a b}$ satisfies

$$R_{a b}=\partial_{a} \phi \partial_{b} \phi,$$

find the energy momentum tensor $T_{a b}$ and use the contracted Bianchi identity to show that, if $\partial_{a} \phi \neq 0$, then

$$\nabla_{a} \nabla^{a} \phi=0$$

Show further that $(*)$ implies

$$\partial_{a}\left(\sqrt{-g} g^{a b} \partial_{b} \phi\right)=0$$