| has gloss | eng: The vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors which are tangent to the fibers. More precisely, if π:E→M is a smooth fiber bundle over a smooth manifold M and e ∈ E with π(e)=x ∈ M, then the vertical space VeE at e is the tangent space Te(Ex) to the fiber Ex containing e. That is, VeE = Te(Eπ(e)). The vertical space is therefore a subspace of TeE, and the union of the vertical spaces is a subbundle VE of TE: this is the vertical bundle of E. |