Vertical bundle

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In mathematics, the vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors which are tangent to the fibers. 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 subspace of T_eE, 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 of the differential dπ:TE→π^-1TM; where π^-1TM is the pullback bundle; symbolically, V_eE=ker(dπ_e). Since dπ_e 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 on E is a choice of a complementary subbundle to VE in TE, called the horizontal bundle of the connection.

[edit] 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. 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.

[edit] References

• Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)). Foundations of Differential Geometry, Vol. 1. Wiley-Interscience. ISBN 0471157333.