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. 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 differential dπ:TE→π*TM identifies the quotient bundle TE/VE with the pullback bundle π*TM. 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.