Vertical Bundle
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High Quality Content by WIKIPEDIA articles! 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 :E M is a smooth fiber bundle over a smooth manifold M and e E with (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(E (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 differenti...
High Quality Content by WIKIPEDIA articles! 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 :E M is a smooth fiber bundle over a smooth manifold M and e E with (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(E (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 d :TE -1TM; where -1TM is the pullback bundle; symbolically, VeE=ker(d e). Since d e is surjective at each point e, it yields a canonical identification of the quotient bundle TE/VE with the pullback -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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