Given a bundle $\pi: E \longrightarrow M$ 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 $ker(d\pi_u:T_u E \rightarrow T_p M)$ and constitute a subbundle of $TE \rightarrow E$ that we denote by $VE\rightarrow E$ 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].
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Author of the notes: Antonio J. Pan-Collantes
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