Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 1 / a / i / Solution

Fix the conventionEquivalently,This defines the Riemann curvature tensor because it is $C^{\infty }(M)$-linear in $U,V,X$: derivatives of a multiplying function cancel between the three terms. Thus its value at a point depends only on the three tangent vectors there, rather than their extensions.