Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 3 / c / Solution

For a path $\gamma :[0,1]\to B$, a section $s(t)$ of ${\gamma }^{\ast }E$ is parallel whenExistence and uniqueness for this linear ordinary differential equation define the parallel transportTransport along the reversed path solves the inverse initial-value problem, so

If $P(t)$ denotes transport from $0$ to $t$, thensatisfies the horizontal equation for the induced endomorphism connection. Uniqueness therefore gives