Vertical and horizontal bundles

In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle. The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle is then a particular choice of a subbundle of the tangent bundle which is complementary to the vertical bundle.

mathematics concept

More precisely, if π : E  M is a smooth fiber bundle over a smooth manifoldM and eE with π(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(Eπ(e)). The vertical space is therefore a vector subspace of TeE. A horizontal space HeE is then a choice of a subspace of TeE such that TeE is the direct sum of VeE and HeE.

The disjoint union of the vertical spaces VeE for each e in E is the subbundle VE of TE: this is the vertical bundle of E. Likewise, a horizontal bundle is the disjoint union of the horizontal subspaces HeE. The use of the words “the” and “a” in this definition is crucial: the vertical subspace is unique, it is determined solely by the fibration. By contrast, there are an infinite number of horizontal subspaces to choose from, in forming the direct sum.

The horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice then becomes equivalent to the definition of a connection on the principal bundle.[1] The choice of a G-invariant horizontal bundle and a connection are the same thing. In the case when E is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, then the structure group G = GLn acts freely and transitively on each fibre, and the choice of a horizontal bundle gives a connection on the frame bundle.

. . . Vertical and horizontal bundles . . .

Let π:EM be a smooth fiber bundle over a smooth manifoldM. The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE  TM.[2]

Since dπe is surjective at each point e, it yields a regular subbundle of TE. Furthermore, the vertical bundle VE is also integrable.

An Ehresmann connection on E is a choice of a complementary subbundle HE to VE in TE, called the horizontal bundle of the connection. At each point e in E, the two subspaces form a direct sum, such that TeE = VeE ⊕ HeE.

A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B1 := (M × N, pr1) with bundle projection pr1 : M×NM : (x, y)  x. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M×N. Then the image of this point under pr1 is m. The preimage of m under this same pr1 is {m} ×N, so that T(m,n) ({m} ×N) = {m} × TN. The vertical bundle is then VB1 = M× TN, 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 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.

. . . Vertical and horizontal bundles . . .

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. . . Vertical and horizontal bundles . . .

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