Vertical bundle

Given a bundle $\pi :E⟶M$ with standard fiber $F$, at each tangent space $T_{u}E$ (considering $u\in E$ and $p=\pi (u)$), we have a vector subspace $V_u$ consisting of those vectors tangent to the fiber $E_p$. They are obtained as $ker(d{\pi }_u:T_{u}E\to T_{p}M)$ and constitute a subbundle of $TE\to E$ that we denote by $VE\to E$ and call the vertical bundle. It is natural, in the sense that we do not have to provide "external information".

However, the introduction of an horizontal bundle is not natural; that would be a connection on a fiber bundle. There is a certain natural bundle, the transversal bundle, which would be like a kind of horizontal bundle but "delocalized" outside of $TE$.

It has a dual version, the horizontal cotangent bundle.

For more information, see [Xournal 117].