(i) Given a covariant vector field $V_{a}$, define the curvature tensor $R_{b c d}^{a}$ by

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

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

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

(ii) Write down an expression similar to (*) given in Part (i) for the quantity

$$g_{a b ; c d}-g_{a b ; d c}$$

and hence show that

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

Define the Ricci tensor, show that it is symmetric and write down the contracted Bianchi identities.

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

$$R_{a b c d}=K\left(x^{e}\right)\left[g_{a c} g_{b d}-g_{a d} g_{b c}\right]$$

Obtain the Ricci tensor and Ricci scalar. Deduce that $K$ is a constant in such spacetimes if the dimension $n$ is greater than 2 .