Let $\Gamma_{b c}^{a}$ be the Levi-Civita connection and $R_{b c d}^{a}$ the Riemann tensor corresponding to a metric $g_{a b}(x)$, and let $\widetilde{\Gamma}_{b c}^{a}$ be the Levi-Civita connection and $\widetilde{R}_{b c d}^{a}$ the Riemann tensor corresponding to a metric $\tilde{g}_{a b}(x)$. Let $T_{b c}^{a}=\widetilde{\Gamma}_{b c}^{a}-\Gamma_{b c}^{a}$.

(a) Show that $T_{b c}^{a}$ is a tensor.

(b) Using local inertial coordinates for the metric $g_{a b}$, or otherwise, show that

$$\widetilde{R}_{b c d}^{a}-R_{b c d}^{a}=2 T_{b[d ; c]}^{a}-2 T_{e[d}^{a} T_{c] b}^{e}$$

holds in all coordinate systems, where the semi-colon denotes covariant differentiation using the connection $\Gamma_{b c}^{a}$. [You may assume that $R_{b c d}^{a}=2 \Gamma_{b[d, c]}^{a}-2 \Gamma_{e[d}^{a} \Gamma_{c] b}^{e}$.]

(c) In the case that $T_{b c}^{a}=\ell^{a} g_{b c}$ for some vector field $\ell^{a}$, show that $\widetilde{R}_{b d}=R_{b d}$ if and only if

$$\ell_{b ; d}+\ell_{b} \ell_{d}=0$$

(d) Using the result that $\ell_{[a ; b]}=0$ if and only if $\ell_{a}=\phi_{, a}$ for some scalar field $\phi$, show that the condition on $\ell_{a}$ in part (c) can be written as

$$k_{a ; b}=0$$

for a certain covector field $k_{a}$, which you should define.