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6 EDUARDO MARTÍNEZBy a distribution along π 10 we mean a subbundle D of 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 of the paper we consider only R-vertical distributions.Theorem. If a distribution along π 10 is parallel then it is integrable.Proof. 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 of 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 of those.It would be nice to understand in more detail the implications of 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 of second-order differential equations that can be decoupled. The followingis an adaptation to the non-autonomous case of the definition of 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 of m fibred over the real line π U : U → R, such that therestriction of 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 expressionof 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 of 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 ),
PARALLEL TRANSPORT AND DECOUPLING 7i.e. the evolution of the x i coordinates is independent of the evolution of thecoordinates x A , v A .Theorem. A sode Γ is submersive if and only if there exists a distribution Dalong π 10 which is parallel and Φ-invariant, that is• D W D ⊂ D for all W ∈ X(J 1 π), and• Φ(D) ⊂ D.Proof. (⇒) Since D is parallel we have that it is involutive. Let (t, x i , x A ), i =1, . . . , d, A = d+1, . . . , n a system of local coordinates, such that D = span{∂/∂x A |A =∂d + 1, . . . , n}. Moreover, since D Γ D ⊂ D we have that D Γ = Γ i ∂x A A ∂∂x+ Γ B i A∂ ∈∂x BD, from where it follows that Γ i A = 0. Thus f i does not depend on the coordinatesv A , i.e. f i = f i (t, x j , x A , v j ).Moreover, D is also Φ-invariant, so that Φ( ∂ ) = Φ i ∂x A A∂∂x+ Φ B i A ∂ ∈ D, form∂x Bwhere we have Φ i A = 0. From the local expression of Φ and taking into accountthat Γ i A = 0 we getΦ i A = − ∂f i∂x A − Γi jΓ j A − Γi BΓ B A − Γ(Γ i A) = − ∂f i∂x A = 0.Thus f i does not depend on x A , v A , i.e. f i = f i (t, x j , v j ). We conclude that thefirst d equations decouple from the others.(⇐) If in some coordinate system (t, x i , x A ) the sode is submersive, then f i =f i (t, x j , v j ) from where we have that ∂f i= 0 and ∂f i= 0. It follows that in∂v A∂x Athis coordinates Γ i A = 0 and Φi A = 0, from where it is clear that the distributionD = 〈 ∂ 〉 is parallel and Φ-invariant.□∂x ADefinition. We say that a sode Γ ∈ X(J 1 π) decouples if there exist two bundlesπ i : E i → R, i = 1, 2, a diffeomorphism ϕ: E → E 1 × R E 2 over the identity in Rand two sodes Γ i on J 1 π i , i = 1, 2 such that Γ and (Γ 1 , Γ 2 ) are J 1 ϕ-related.We say that the sode Γ locally decouples at a point m ∈ E if there is an openneighbourhood U ⊂ E of m fibred over the real line π U : U → R, such that therestriction of Γ to J 1 π U decouples.It is clear that Γ decouples if and only if it is submersive with respect to twocomplementary subbundles E 1 and E 2 , where by complementary we mean thatE 1 × R E 2 is (diffeomorphic to) E. Therefore, in adapted coordinates (t, x i , x A ), wehave that the differential equations for the integral curves of Γ are the union of twoseparate subsystem of second-order differential equations:ẍ i = f i (t, x j , v j ) ẍ A = f A (t, x A , v A ),From the above observation and the results in this section we immediately getthe following result, where as above the word ‘decouples’ must be understood as‘locally decouples’.Theorem. Let D 1 , D 2 be two complementary distributions, in the sense that, π ∗ 10(Ver(π)) =D 1 ⊕ D 2 , with dimension d 1 and d 2 , respectively (and d 1 + d 2 = n). If both distributionsare parallel and Φ-invariant, then the second-order differential equationdecouples in two subsystems of dimension d 1 and d 2 .And iterating the above process
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