Vertical and horizontal bundles

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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 tangent bundle which is complementary to vertical bundle.

More precisely, if π : E → M is a smooth fiber bundle over a smooth manifold M 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 principle 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.

Formal definition[edit]

Let π:EM be a smooth fiber bundle over a smooth manifold M. 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.

Example[edit]

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[edit]

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:

  • A vertical vector field is a vector field that is in the vertical bundle. That is, for each point e of E, one chooses a vector where is the vertical vector space at e.[2]
  • A differentiable r-form on E is said to be a horizontal form if whenever at least one of the vectors is vertical.
  • The connection form vanishes on the horizontal bundle, and is non-zero only on the vertical bundle. In this way, the connection form can be used to define the horizontal bundle: The horizontal bundle is the kernel of the connection form.
  • The solder form or tautological one-form vanishes on the vertical bundle and is non-zero only on the horizontal bundle. By definition, the solder form takes its values entirely in the vertical bundle.
  • For the case of a frame bundle, the torsion form vanishes on the vertical bundle, and can be used to define exactly that part that needs to be added to an arbitrary connection to turn it into a Levi-Civita connection, i.e. to make a connection be torsionless. Indeed, if one writes θ for the solder form, then the torsion tensor Θ is given by Θ = D θ (with D the exterior covariant derivative). For any given connection ω, there is a unique one-form σ on TE, called the contorsion tensor, that is vanishing in the vertical bundle, and is such that ω+σ is another connection 1-form that is torsion-free. The resulting one-form ω+σ is nothing other than the Levi-Civita connection. One can take this as a definition: since the torsion is given by , the vanishing of the torsion is equivalent to having , and it is not hard to show that σ must vanish on the vertical bundle, and that σ must be G-invariant on each fibre (more precisely, that σ transforms in the adjoint representation of G). Note that this defines the Levi-Civita connection without making any explicit reference to any metric tensor (although the metric tensor can be understood to be a special case of a solder form, as it establishes a mapping between the tangent and cotangent bundles of the base space, i.e. between the horizontal and vertical subspaces of the frame bundle).
  • In the case where E is a principle bundle, then the fundamental vector field must necessarily live in the vertical bundle, and vanish in any horizontal bundle.

Notes[edit]

  1. ^ David Bleeker, Gauge Theory and Variational Principles (1991) 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[edit]