Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 38E / b / Solution

Set $U_{\mu }={\partial }_{\mu }\varphi$. Torsion-freeness and commutation of ordinary partial derivatives imply that its covariant Hessian is symmetric:Take the cyclic sum of the three commutators acting respectively on $U_{\gamma },U_{\alpha },U_{\beta }$. Their left-hand sides cancel pairwise by this Hessian symmetry, leavingAt a chosen point, a scalar field can be selected whose gradient has any prescribed covector value. The coefficient of $U_{\nu }$ must therefore vanish. Relabelling indices gives the first Bianchi identity