Vertical space

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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_pM \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.