Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 4 / c / Solution

WriteFor a diagonal curvature operator, ${\Theta }^{\mu \nu }$ contains only $f^{\mu }\wedge f^{\nu }$. Consequently $R^{\mu }{}_{\nu \rho \sigma }$ can be nonzero only when the plane indexed by $\rho ,\sigma$ is the same as the plane indexed by $\mu ,\nu$. In the contractionthis condition cannot hold for $\nu \ne \sigma$. Thereforewhich is the statement that a diagonal curvature operator implies diagonal Ricci tensor.