Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 309 / 4 / c / Solution

Using Cartan's second structure equation givesSince ${\Theta }_{\mu \nu }=R_{\mu \nu \rho \sigma }{e}^{\rho }\wedge e^{\sigma }/2$, the independent nonzero lowered Riemann curvature tensor components arewith all others determined by the symmetries of the Riemann curvature tensor.