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.

More precisely, if π : E → M is a smooth fiber bundle over a smooth manifold M and e ∈ E with π(e) = x ∈ M, then the vertical space V_eE at e is the tangent space T_e(E_x) to the fiber E_x containing e. That is, V_eE = T_e(E_π(e)). The vertical space is therefore a vector subspace of T_eE. A horizontal space H_eE is then a choice of a subspace of T_eE such that T_eE is the direct sum of V_eE and H_eE.

The disjoint union of the vertical spaces V_eE 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 H_eE. 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 = GL_n acts freely and transitively on each fibre, and the choice of a horizontal bundle gives a connection on the frame bundle.

Formal definition

Let π:E→M 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 dπ_e 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 T_eE = V_eE ⊕ H_eE.

Example

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 × N → M : (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.

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:

• 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 ${\displaystyle v_{e}\in V_{e}E}$ where ${\displaystyle V_{e}E\subset T_{e}E=T_{e}(E_{\pi (e)})}$ is the vertical vector space at e.[2]
• A differentiable r-form ${\displaystyle \alpha }$ on E is said to be a horizontal form if ${\displaystyle \alpha (v_{1},...,v_{r})=0}$ whenever at least one of the vectors ${\displaystyle v_{1},...,v_{r}}$ 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 horizontal 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 ${\displaystyle \Theta =D\theta =d\theta +\omega \wedge \theta }$, the vanishing of the torsion is equivalent to having ${\displaystyle d\theta =-(\omega +\sigma )\wedge \theta }$, 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 principal bundle, then the fundamental vector field must necessarily live in the vertical bundle, and vanish in any horizontal bundle.

Notes

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

References

• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley Interscience. ISBN 0-471-15733-3.
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural Operations in Differential Geometry (PDF), Springer-Verlag
• Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
• Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7