Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 3 / a / Solution

For $\pi :E\to B$, the vertical space at $p\in E$ is $V_{p}E=\ker (d\pi {)}_p$. A horizontal subspace is a complementary subspaceA connection is a smooth choice of such complements, linear with respect to the vector-bundle structure. Equivalently, it is a smooth horizontal distribution as in horizontal subspace of a vector bundle connection.