In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle , the vertical bundle and horizontal bundle are subbundles of the tangent bundle of whose Whitney sum satisfies . This means that, over each point , the fibers and form complementary subspaces of the tangent space . The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle requires some choice of complementary subbundle.

To make this precise, define the vertical space at to be . That is, the differential (where ) is a linear surjection whose kernel has the same dimension as the fibers of . If we write , then consists of exactly the vectors in which are also tangent to . The name is motivated by low-dimensional examples like the trivial line bundle over a circle, which is sometimes depicted as a vertical cylinder projecting to a horizontal circle. A subspace of is called a horizontal space if is the direct sum of and .

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, provided the horizontal spaces vary smoothly with e, their disjoint union is a horizontal bundle. The use of the words "the" and "a" here is intentional: each vertical subspace is unique, defined explicitly by . Excluding trivial cases, there are an infinite number of horizontal subspaces at each point. Also note that arbitrary choices of horizontal space at each point will not, in general, form a smooth vector bundle; they must also vary in an appropriately smooth way.

The horizontal bundle 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 is equivalent to a connection on the principal bundle.[1] This notably occurs when E is the frame bundle associated to some vector bundle, which is a principal bundle.

Formal definition

Let π:EB be a smooth fiber bundle over a smooth manifold B. The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TB.[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.

Example

The Möbius strip is a line bundle over the circle, and the circle can be pictured as the middle ring of the strip. At each point on the strip, the projection map projects it towards the middle ring, and the fiber is perpendicular to the middle ring. The vertical bundle at this point is the tangent space to the fiber.

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 : (xy) → 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 : (xy) → 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.

Properties

Various important tensors and differential forms from differential geometry take on specific properties on the vertical and horizontal bundles, or even can be defined in terms of them. Some of these are:

Notes

  1. ^ David Bleecker, Gauge Theory and Variational Principles (1981) Addison-Wesely Publishing Company ISBN 0-201-10096-7 (See theorem 1.2.4)
  2. ^ a b Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural Operations in Differential Geometry (PDF), Springer-Verlag (page 77)

References