Vertical space

Definition

Let $f:M\to N$ be a smooth map of differential manifolds. Given a point $m\in M$, the vertical space at $m$ is the kernel of the map $(Df{)}_p:T_{p}M\to T_{f(p)}(N)$.

A vector that lies in the vertical space at $p$ is termed a vertical vector at $p$.

We typically use the notions of vertical space and vertical vector for a bundle map, where we are thinking of the base as sitting horizontally, and the fibers as sitting vertically. Then, a vertical vector is a vector that is literally vertical, and the vertical space is the vertical subspace of te tangent space.

There is no natural notion of a horizontal space in general. However, when we have a principal bundle, and if we provide a connection on it, then we do get a notion of horizontal space at each point, and this is a direct sum complement to the vertical space.