Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 2 / a / i / Solution

Fix the conventionAlthough a single covariant derivative of a vector is tensorial, the second derivative contains connection-dependent terms. Their antisymmetric difference cancels every second derivative of a coordinate change. More intrinsically, the mapis $C^{\infty }(M)$-linear in each of $U,V,X$, so it defines the Riemann curvature tensor.