PARALLEL TRANSPORT AND DECOUPLING 1. Introduction One of ...
PARALLEL TRANSPORT AND DECOUPLING 1. Introduction One of ...
PARALLEL TRANSPORT AND DECOUPLING 1. Introduction One of ...
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6 EDUARDO MARTÍNEZBy a distribution along π 10 we mean a subbundle D <strong>of</strong> π ∗ 10(T E). A distributionalong π 10 is said to be basic if there exists a distribution E on E such that D =E ◦ π 10 , that is, D v = E π10(v) for every v ∈ J 1 π. A distribution D along π 10 is saidto be integrable if it is basic D = E ◦ π 10 and the distribution E is integrable.On the other hand we will say that a distribution D along π 10 is parallel if it isinvariant under parallel transport P γ D ⊂ D for every curve γ in J 1 π. It is easy tosee that D is parallel if and only if it is invariant by covariant differentiation, i.e.D W D ⊂ D for all W ∈ X(J 1 π).As we mentioned before, we only consider the subbundle π ∗ 10(Ver(π)) ⊂ π ∗ 10(T E),and therefore in the rest <strong>of</strong> the paper we consider only R-vertical distributions.Theorem. If a distribution along π 10 is parallel then it is integrable.Pro<strong>of</strong>. If D is invariant by parallel transport over vertical curves then it is a basicdistribution. Indeed, parallel transport along vertical curves is (p i , z) ↦→ (p f , z),and therefore, a distribution D is parallel along vertical curves iff D p depends onlyon the point m = π 10 (p). Defining E m = D 0m then we have that D p = E m .Moreover, since the connection is torsionless we have that D X H Y − D Y H X =[X, Y ] for all X, Y ∈ X(E). Therefore if X, Y are vectorfields in the distribution Ewe have that D X H Y and D X H Y are in E, and hence [X, Y ] is also in E. Thus E isinvolutive and therefore integrable.□It is to be noticed that from the properties <strong>of</strong> the linear connection D, one cansee that it is enough to impose the invariance condition D Z D ⊂ D for π 10 -verticalvector fields Z and for Z = Γ, being the invariance under D Z for Z a horizontalvectorfield a consequence <strong>of</strong> those.It would be nice to understand in more detail the implications <strong>of</strong> the torsionfreecondition in what respect to parallel transport along horizontal curves.6. Submersive sodesIn this section we will use the results in the last section in order to characterizethose systems <strong>of</strong> second-order differential equations that can be decoupled. The followingis an adaptation to the non-autonomous case <strong>of</strong> the definition <strong>of</strong> submersivesode given in [7] for the time independent case.Definition. A sode Γ ∈ X(J 1 π) is submersive if there exists a bundle ¯π : Ē → R,a submersion ϕ: E → Ē over the identity in R, and a sode ¯Γ on J 1¯π such that Γand ¯Γ are J 1 ϕ-related, i.e. T (J 1 ϕ) ◦ Γ = ¯Γ ◦ J 1 ϕ.We say that a sode Γ is locally submersive at a point m ∈ E if there is anopen neighbourhood U ⊂ E <strong>of</strong> m fibred over the real line π U : U → R, such that therestriction <strong>of</strong> Γ to J 1 π U is submersive.We will only consider the local problem, i.e. the word ‘submersive’ must beunderstood as ‘locally submersive’.We can take coordinates adapted to the submersion ϕ, i.e. (t, x i ) on Ē and(t, x i , x A ) on E, with i = 1, . . . , k, A = k+1, . . . , n, such that the coordinate expression<strong>of</strong> ϕ is ϕ(t, x i , x A ) = (t, x i ). Then the sode Γ with forces f i (t, x j , x B , v j , v B )and f A (t, x j , x B , v j , v B ) is submersive iff the coefficients f i depends only <strong>of</strong> the coordinates(t, x i , v i ) and does not depend on (x B , v B ). It follows that the differentialequations for the integral curves can be written asẍ i = f i (t, x j , v j ) ẍ A = f A (t, x j , x A , v j , v A ),