Vertical Bundle

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 : EM is a smooth fiber bundle over a smooth manifold M and eE with π(e) = xM, 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 : TEπ−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.

Read more about Vertical Bundle:  Example

Famous quotes containing the words vertical and/or bundle:

    In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.
    Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)

    In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.
    Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)