Starting from the Riemann tensor for a metric $g_{a b}$, define the Ricci tensor $R_{a b}$ and the scalar curvature $R$.

The Riemann tensor obeys

$$\nabla_{e} R_{a b c d}+\nabla_{c} R_{a b d e}+\nabla_{d} R_{a b e c}=0$$

Deduce that

$$\nabla^{a} R_{a b}=\frac{1}{2} \nabla_{b} R$$

Write down Einstein's field equations in the presence of a matter source, with energymomentum tensor $T_{a b}$. How is the relation $(*)$ important for the consistency of Einstein's equations?

Show that, for a scalar function $\phi$, one has

$$\nabla^{2} \nabla_{a} \phi=\nabla_{a} \nabla^{2} \phi+R_{a b} \nabla^{b} \phi .$$

Assume that

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

for a scalar field $\phi$. Show that the quantity

$$R+\nabla^{a} \phi \nabla_{a} \phi$$

is then a constant.