Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 309 / 4 / i / Solution

Use $t$ as an affine parameter on each member of the geodesic variation, so ${\nabla }_{T}T=0$. The parameter directions commute, giving $[T,S]=0$ along the variation. The torsion-free connection then givesConsequently the covariant derivative and curvature definition giveThis is the geodesic-deviation equation with the stated curvature sign:The component expression defines the slot order: $\beta$ is the vector acted on, and $\gamma ,\delta$ are the two derivative directions of the Riemann curvature tensor. Both the commuting parameter directions and affine parametrization are essential; a general connecting vector field or a nonaffine parameter would add terms.