Horizontal bundle

Horizontal bundle

In mathematics, in the field of differential topology, given

:"π":"E"→"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 tangent to the fibers of "E" over "M". A horizontal bundle is then a particular choice of a subbundle of T"E" which is complementary to V"E", in other words provides a complementary subspace in each fiber.

In full generality, the horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle. However, the concept is usually applied in more specific contexts.

More precisely, if "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" through "e". A horizontal bundle then determines an horizontal space H"e""E" such that T"e""E" is the direct sum of V"e""E" and H"e""E".

If "E" is a principal "G"-bundle then the horizontal bundle is usually required to be "G"-invariant: see Connection (principal bundle) for further details. In particular, this is the case when "E" is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, and "G" = GL"n".


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • 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

  • 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 pi; : E rarr; M is a smooth fiber bundle over a smooth manifold M …   Wikipedia

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

  • Frame bundle — In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex. The general linear group acts naturally on F(E) via a change… …   Wikipedia

  • Banach bundle — In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.Definition of a Banach bundleLet M be a Banach manifold of class C p with p ≥ 0, called …   Wikipedia

  • Adjoint bundle — In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into an algebra bundle. Adjoint bundles has important… …   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

  • 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

  • List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

  • Linear connection — In the mathematical field of differential geometry, the term linear connection can refer to either of the following overlapping concepts: * a connection on a vector bundle, often viewed as a differential operator (a Koszul connection or covariant …   Wikipedia

Share the article and excerpts

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