Vertical bundle

Vertical bundle

In mathematics, 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 V"e""E" at "e" is the tangent space T"e"("E""x") to the fiber "E""x" containing "e". That is, V"e""E" = T"e"(E"π"("e")). The vertical space is therefore a subspace of T"e""E", and the union of the vertical spaces is a subbundle V"E" of T"E": this is the vertical bundle of "E".

The vertical bundle is the kernel of the differential d"π":T"E"→"π"-1T"M"; where π-1T"M" is the pullback bundle; symbolically, Ve"E"=ker(dπe). Since dπe is surjective at each point "e", it yields a canonical identification of the quotient bundle T"E"/V"E" with the pullback "π"-1T"M".

An Ehresmann connection on "E" is a choice of a complementary subbundle to V"E" in T"E", called the horizontal bundle of the connection.

Example

A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle "B"1 := ("M" × "N", pr1) with bundle projection pr1 : "M" × "N" → "M" : ("x", "y") → "x". The vertical bundle is then V"B"1 = "M" × T"N", which is a subbundle of T("M"×"N"). If we take the other projection pr2 : "M" × "N" → "N" : ("x", "y") → "y" to define the fiber bundle "B"2 := ("M" × "N", pr2) then the vertical bundle will be V"B"2 = T"M" × "N".

In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of "B"1 is the vertical bundle of "B"2 and vice versa.

References

*


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Vertical agreement — A vertical agreement is a term used in competition law to denote agreements between firms up or down the supply chain from one another. For instance, between a manufacturer of electronic gadgets like Sony might have a vertical agreement with and… …   Wikipedia

  • Connection (principal bundle) — This article is about connections on principal bundles. See connection (mathematics) for other types of connections in mathematics. In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way …   Wikipedia

  • Horizontal bundle — In mathematics, in the field of differential topology, given : pi; : E rarr; M , a smooth fiber bundle over a smooth manifold M , then the vertical bundle V E of E is the subbundle of the tangent bundle T E consisting of the vectors which are… …   Wikipedia

  • Double tangent bundle — In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM,M) of a smooth manifold M [1]. The second… …   Wikipedia

  • Vector bundle — The Möbius strip is a line bundle over the 1 sphere S1. Locally around every point in S1, it looks like U × R, but the total bundle is different from S1 × R (which is a cylinder instead). In mathematics, a vector bundle is a… …   Wikipedia

  • Fiber bundle — In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which looks locally like a product space. It may have a different global topological structure in that the space as a whole may not be homeomorphic to a… …   Wikipedia

  • Tangent bundle — In mathematics, the tangent bundle of a smooth (or differentiable) manifold M , denoted by T ( M ) or just TM , is the disjoint unionThe disjoint union assures that for any two points x 1 and x 2 of manifold M the tangent spaces T 1 and T 2 have… …   Wikipedia

  • Ehresmann connection — In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection which is defined on arbitrary fibre bundles. In particular, it may… …   Wikipedia

  • Solder form — In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fibre bundle to a smooth manifold is a manner of attaching the fibres to the manifold in such a way that they can be regarded as tangent.… …   Wikipedia

  • Connexion de Ehresmann — En géométrie différentielle, une connexion de Ehresmann (d après le mathématicien français Charles Ehresmann qui a le premier formalisé ce concept) est une version de la notion de connexion qui est définie sur des fibrés. En particulier, elle… …   Wikipédia en Français

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”