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.
The vertical bundle is the kernel of the differential dπ : TE → π−1TM; where π−1TM is the pullback bundle; symbolically, VeE = ker(dπe). Since dπe is surjective at each point e, it yields a canonical identification of the quotient bundle TE/VE with the pullback π−1TM.
An Ehresmann connection on E is a choice of a complementary subbundle to VE in TE, called the horizontal bundle of the connection.
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