Let $(\mathcal{M}, \boldsymbol{g})$ be a spacetime and $\boldsymbol{\Gamma}$ the Levi-Civita connection of the metric $\boldsymbol{g}$. The Riemann tensor of this spacetime is given in terms of the connection by

$$R_{\rho \alpha \beta}^{\gamma}=\partial_{\alpha} \Gamma_{\rho \beta}^{\gamma}-\partial_{\beta} \Gamma_{\rho \alpha}^{\gamma}+\Gamma_{\rho \beta}^{\mu} \Gamma_{\mu \alpha}^{\gamma}-\Gamma_{\rho \alpha}^{\mu} \Gamma_{\mu \beta}^{\gamma}$$

The contracted Bianchi identities ensure that the Einstein tensor satisfies

$$\nabla^{\mu} G_{\mu \nu}=0$$

(a) Show that the Riemann tensor obeys the symmetry

$$R_{\rho \alpha \beta}^{\mu}+R_{\beta \rho \alpha}^{\mu}+R_{\alpha \beta \rho}^{\mu}=0 .$$

(b) Show that a vector field $V^{\alpha}$ satisfies the Ricci identity

$$2 \nabla_{[\alpha} \nabla_{\beta]} V^{\gamma}=\nabla_{\alpha} \nabla_{\beta} V^{\gamma}-\nabla_{\beta} \nabla_{\alpha} V^{\gamma}=R_{\rho \alpha \beta}^{\gamma} V^{\rho}$$

Calculate the analogous expression for a rank $\left(\begin{array}{l}2 \\ 0\end{array}\right)$ tensor $T^{\mu \nu}$, i.e. calculate $\nabla_{[\alpha} \nabla_{\beta]} T^{\mu \nu}$ in terms of the Riemann tensor.

(c) Let $K^{\alpha}$ be a vector that satisfies the Killing equation

$$\nabla_{\alpha} K_{\beta}+\nabla_{\beta} K_{\alpha}=0$$

Use the symmetry relation of part (a) to show that

$$\begin{gathered} \nabla_{\nu} \nabla_{\mu} K^{\alpha}=R_{\mu \nu \beta}^{\alpha} K^{\beta} \\ \nabla^{\mu} \nabla_{\mu} K^{\alpha}=-R_{\beta}^{\alpha} K^{\beta} \end{gathered}$$

where $R_{\alpha \beta}$ is the Ricci tensor.

(d) Show that

$$K^{\alpha} \nabla_{\alpha} R=2 \nabla^{[\mu} \nabla^{\lambda]} \nabla_{[\mu} K_{\lambda]},$$

and use the result of part (b) to show that the right hand side evaluates to zero, hence showing that $K^{\alpha} \nabla_{\alpha} R=0$.