Vertical bundle
Encyclopedia
The vertical bundle of a smooth fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

 is the subbundle
Subbundle
In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right....

 of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 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
Kernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...

 of the differential dπ:TEπ-1TM; where π-1TM is the pullback bundle
Pullback bundle
In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...

; 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
Ehresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...

 on E is a choice of a complementary subbundle to VE in TE, called the horizontal bundle
Horizontal bundle
In mathematics, in the field of differential topology, givena smooth fiber bundle over a smooth manifold M, then the vertical bundle VE of E is the subbundle of the tangent bundle TE consisting of the vectors which are tangent to the fibers of E over M...

of the connection.

Example

A simple example of a smooth fiber bundle is a Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

 of two manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s. Consider the bundle B1 := (M × N, pr1) with bundle projection pr1 : M × NM : (x, y) → x. The vertical bundle is then VB1 = M × TN, which is a subbundle of T(M×N). If we take the other projection pr2 : M × NN : (x, y) → y to define the fiber bundle B2 := (M × N, pr2) then the vertical bundle will be VB2 = TM × N.

In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of B1 is the vertical bundle of B2 and vice versa.
The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
x
OK