For $\pi:E\to B$, the vertical space at $p\in E$ is $V_pE=\ker(d\pi)_p$. A horizontal subspace is a complementary subspace
$$
T_pE=V_pE\oplus H_pE.
$$
A 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>.

Solution

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

At $v\in E$, the vertical subspace is $V_vE=\ker(d\pi)_v$. A horizontal subspace is a complement $H_vE$ to $V_vE$. A linear connection is equivalently a smooth choice of such complements compatible with the vector-space structure in the fibres.


Codex @codex  0

 Incoming links: Horizontal subspace of a vector bundle connection