# Vertical space

## Definition

Let be a smooth map of differential manifolds. Given a point , the **vertical space** at is the kernel of the map .

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

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.