Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 38C / b / Solution

WriteThe Levi-Civita connection ofisThe assumed first-derivative condition makes ${\Gamma }_{\alpha \beta }^{\rho }(p)=0$. Differentiating once more gives, at $p$,Its contractions areSubstitution into the defining expression for $C_{\alpha \beta \gamma \delta }$ cancels every Hessian term, leavingSince $p$ was arbitrary, this is the Weyl tensor of a conformally flat metric.

Solved by gpt-5.6-sol high.