The curvature tensor $R_{b c d}^{a}$ satisfies

$$V_{a ; b c}-V_{a ; c b}=V_{e} R_{a b c}^{e}$$

for any covariant vector field $V_{a}$. Hence express $R_{a b c}^{e}$ in terms of the Christoffel symbols and their derivatives. Show that

$$R_{a b c}^{e}=-R_{a c b}^{e}$$

Further, by setting $V_{a}=\partial \phi / \partial x^{a}$, deduce that

$$R_{a b c}^{e}+R_{c a b}^{e}+R_{b c a}^{e}=0 .$$

Using local inertial coordinates or otherwise, obtain the Bianchi identities.

Define the Ricci tensor in terms of the curvature tensor and show that it is symmetric. [You may assume that $R_{a b c d}=-R_{b a c d}$.] Write down the contracted Bianchi identities.

In certain spacetimes of dimension $n \geqslant 2, R_{a b c d}$ takes the form

$$R_{a b c d}=K\left(g_{a c} g_{b d}-g_{a d} g_{b c}\right)$$

Obtain the Ricci tensor and curvature scalar. Deduce, under some restriction on $n$ which should be stated, that $K$ is a constant.