PARALLEL TRANSPORT AND DECOUPLING 1. Introduction One of ...

PARALLEL TRANSPORT AND DECOUPLINGEDUARDO MARTÍNEZAbstract. The problem of decoupling second-order differential equations bycoordinate transformations is studied in terms of parallel distributions withrespect to the linear connection associated to that differential equation.1. IntroductionOne of the most fruitful tools in the analysis of qualitative properties of systemof second-order differential equations (sode) during the last decades has been thenonlinear connection [2] associated to such sode and the corresponding linear connection[5], also known as Berwald connection. For instance, it is very useful in theanalysis of the Inverse Problem of Lagrangian Mechanics [11], it is fundamental tocharacterize those sodes which can be transformed to linear sodes by a coordinatetransformation [9, 5] and also to characterize those systems of second-orderdifferential equations which decouples under a coordinate transformation in twosubsystems of independent second-order differential equations [8, 1], to mention afew problems in which it has been successfully applied.In [6] Crampin provides a geometrical interpretation of the parallel transportmap associated to the linear connection D along horizontal and vertical curves,which moreover characterizes such connection. The interpretation of the paralleltransport along integral curves of a projectable horizontal vectorfield is given interms of Lie transport under the flow of this vectorfield. This fact strongly suggeststhat a parallel distribution must satisfy some property of invariance under someflows, and therefore there should be a connection with the integrability of suchdistribution.The aim of this paper is to explore this relation and to apply this results to theproblem of decoupling of second-order differential equations, providing a simplifiedalternative derivation of the results in [1]. In this sense many of the results in thispaper are already in the literature, and its merit is mainly to provide simpler andmore readable proofs of such results.The paper is organized as follows. In section 2 the notation and preliminaryresults that will be needed in the rest of the paper are introduced. In section 3 werecall the construction of the linear connection associated to the nonlinear connectiondefined by a sode. In section 4 we recall Crampin’s interpretation of paralleltransport for such connection, and in section 5 we show that a parallel subbundleis necessarily integrable. In section 6 we characterize those sodes which can be2000 Mathematics Subject Classification. 34A26,34C35,53B05.Key words and phrases. Second-order differential equations, linear connections, nonlinear connections,parallel transport, decoupling of differential equations.I would like to thank to Tom Mestadg and Willy Sarlet for many illuminating discussions.1

2 EDUARDO MARTÍNEZdecoupled by a coordinate change and finally in section 7 the case of a completedecoupling into scalar second-order differential equations is analyzed.2. PreliminariesLet π : E → R be a fibre bundle with fibre dimension n, and let π 1 : J 1 π → Rbe its first jet bundle. The vertical bundle with respect to the bundle projectionπ is denoted by Ver(π), whereas the vertical bundle with respect to the projectionπ 10 : J 1 π → E will be denoted by Ver(π 10 ), i.e. Ver(π) = Ker(T π) and Ver(π 10 ) =Ker(T π 10 ).We consider the canonical coordinate t on R and natural bundle coordinates(t, x i ) on E and (t, x i , v i ) on J 1 π. Any time-preserving coordinate transformation(t, x i ) → (t, ¯x i ), where ¯x i = ¯x i (t, x) leads to the following formulas for thecoordinate transformation on J 1 π,t = t, ¯x i = ¯x i (t, x), ¯v i = ∂¯xi∂t + ∂¯xi∂x j vj ,from where we can clearly see the affine character of the bundle J 1 π, whoseassociated vector bundle is Ver(π). The fibre over an element m ∈ E can beconsidered as an affine hyperplane of the tangent space at m, and therefore wehave the following sequence of vector spaces 0 → Ver(π) m → T m E → R → 0, wherethe map in the right consists in taking the t-component v ↦→ 〈dt, v〉. Elements ofJ 1 π can be identified with tangent vectors to E which projects onto ∂/∂t. Thisidentification may be regarded as defining a map T : J 1 π → T E, given by T (jt 1 γ) =˙γ(t). We therefore have the following commutative diagram of vector bundles overE, where the row is an exact sequence,0 Ver(π)i T Ej E × R 0.T(π 10,1) J 1 πA section of π ∗ 10(T E) is said to be a vectorfield along π 10 . Alternatively, it canbe considered as a map X : J 1 π → T E such that τ E ◦X = π 10 , that is X(p) ∈ T m Efor every p ∈ (J 1 π) m . The section associated to this map is just p ↦→ (p, X(p)).For instance, the map T above can be considered in a natural way as a vector fieldalong π 10 , called the total time derivative, and which in coordinates readsT = ∂ ∂t + ∂vi∂x i .Any vectorfield Y on E gives rise to a vectorfield along π 10 by composition withthe projection π 10 . Explicitly the associated section of the pullback bundle is p ↦→(p, Y (m)) where m = π 10 (p). The vectorfields along π 10 which arise in this way arecalled basic.A sode on E is a vectorfield Γ ∈ X(J 1 π) which projects onto T . In coordinatesit is of the formΓ = ∂ ∂t + ∂vi∂x i + f i ∂∂v iwhere f i = f i (t, x j , v j ). Therefore the system of differential equations for the integralcurves of Γ is the non-autonomous second-order system of differential equations,

PARALLEL TRANSPORT AND DECOUPLING 3written as a first order system,ẋ i = v i˙v i = f i (t, x, v).or in second-order form ẍ i = f i (t, x, ẋ).After a time-dependent coordinate transformation the coordinate expression forΓ becomesΓ = ∂ ∂t + ∂¯vi ∂ȳ i + ¯f i ∂∂¯v iwhere¯f i = ∂¯xi∂x j f j +∂2¯x i∂x j ∂x k vj v k + 2 ∂2¯x i∂x j ∂t vj + ∂2¯x i∂t 2 .a formula which remind us how the fiber coordinates v i enter into the transformationlaw.The problem we are faced to is to find a coordinate transformation such thatthe system of second-order differential equations is transformed to the so-calledsubmersive form { ẍi = f i (t, x j , ẋ j )i, j = 1, . . . , dẍ A = f A (t, x j , x B , ẋ j , ẋ B )A, B = d + 1, . . . , nor to a separated or decoupled form{ ẍi = f i (t, x j , ẋ j ) i, j = 1, . . . , dẍ A = f A (t, x B , ẋ B ) A, B = d + 1, . . . , n.3. The linear connection associated to a sodeThe fibration π 10 : J 1 π → E determines an exact sequence of vector bundles overJ 1 π0 π ∗ 10 Ver(π)ξ V T (J 1 π)j π ∗ 10(T E) 0 ,where ξ V is the vertical lifting map (given by ξ V (p, v) = d dt | t=0(p + tv)) and j isthe projection j(V p ) = (p, T π 10 (V p )).A (nonlinear) connection on J 1 π is a splitting of such sequence, that is, a linearbundle map h: π10(T ∗ E) → T (J 1 π) such that j ◦ h = id. In other words, it isequivalent to a vector subbundle Hor(π 10 ) of T (J 1 π) which is complementary tothe vertical subbundle Ver(π 10 ), and is therefore called horizontal.A sode Γ on J 1 π determines a connection on J 1 π. The projector onto thehorizontal bundle is given in terms of the vertical endomorphism S = (dx i −v i dt)⊗(∂/∂v i ) by the formulaP H = 1 2 (I + Γ ⊗ dt − L ΓS) .The coordinate expressions for a basis {H 0 , H 1 , . . . , H n } of horizontal vector fieldsareH 0 = ∂ ∂t − ∂Γj 0∂v j and H i = ∂∂x i − ∂Γj i∂v jwhereΓ j 0 = −f i + 1 i∂fvj2 ∂v j and Γ j i = −1 ∂f j2 ∂v i .Note that, in particular, the sode Γ is itself horizontal; it is the horizontal liftof the canonical vectorfield T .

4 EDUARDO MARTÍNEZBy a kind of linearization of the given nonlinear connection, we can define a linearconnection on the pullback bundle pr 1 : π ∗ 10T E → J 1 π. The associated covariantderivative is given byD Z X = κ([P H Z, Y V ]) + j([P V Z, Y H ]) + P H (Z)〈Y, dt〉T ,where Z is a vectorfield on J 1 π, Y is a vectorfield along π 10 (i.e. a section ofπ10T ∗ E), P H and P V are the horizontal and vertical projectors of the nonlinearconnection, Y H and Y V are the horizontal and vertical lifting of Y , and κ is theconnection map κ: T (J 1 π) → π10(Ver(π)), ∗ defined by the relation ξ V ◦ κ = P V .See [5, 4] for the details.The curvature of this connection was also studied in [5]. In particular, most ofthe components of the curvature tensor are determined in terms of the so calledJacobi endomorphism, Φ defined byΦ(X) = R(T , X),where R is the curvature of the nonlinear connection. Its coordinate expression isΦ = Φ i j (dxj − v j dt) ⊗ ∂∂qwithiΦ i j = − ∂f i∂x j − Γi kΓ k j − Γ(Γ i j).We remark that in general, for any connection on J 1 π, the horizontal lift of T isa sode on J 1 π, but it is to be noticed that the connection defined by this sode isnot the original one. This is the case only when the connection is a sode-connectionas above. Notice also that not every connection is the sode-connection for somesode.Proposition. A linear connection on π ∗ 10(T E) is the linear connection defined bysode if and only if it is torsionless, i.e.D X H Y − D Y H X = [X, Y ]for every pair of basic vectorfields X, Y ∈ X(E).The torsionless condition can be equivalently written in the form [X H , Y V ] −[Y H , X V ] = [X, Y ] V for every pair of basic vectorfields X, Y .In the local base {Γ, H i , V i } of vectorfields in J 1 π, where V i = ∂/∂v i , and thelocal base {T , ∂/∂x i } of vectofields along π 10 the linear connection is determinedby( ) ∂D Vi∂x j = 0, D Vi T = ∂∂x i ,D Hi( ∂∂x j )= ∂Γk i∂v j∂∂x k , D H iT = 0,( ) ∂D Γ∂x j = Γ k ∂j∂x k , D ΓT = 0.In particular, from this expressions it is clear that D restricts to a connection onπ ∗ 10(Ver(π)): if Y is vertical over R, then the expression of the covariant derivativesimplifies toD Z X = κ([P H Z, Y V ]) + j([P V Z, Y H ]).

PARALLEL TRANSPORT AND DECOUPLING 5In what follows, to simplify as much as possible, we will consider the restriction ofthe connection to π ∗ 10(Ver(π)). Therefore, we will consider only vertical vectorfieldsalong π 10 .4. Parallel transportThe parallel transport map associated to the linear connection defined abovewas interpreted in the case of the autonomous formalism by Crampin in [6], andextended to the non-autonomous case in [10]. In this section this interpretation ispresented (in a slightly different way).Let X be a vector field on E and X H ∈ X(J 1 π) its horizontal lift with respectto the given connection. Denote by ϕ t the flow of X and by φ t the flow of X H .When the connection is linear, then φ t is a linear map in the tangent bundle T E,which is but the parallel transport map along the integral curves of X. Whenthe connection is nonlinear, we can linearize the flow obtaining a linear bundlemap Ψ t : π10(Ver(π)) ∗ → π10(Ver(π)) ∗ over φ t as follows. We take p ∈ J 1 π andv ∈ Ver(π), over the same point m ∈ E, so that (p, v) ∈ π10(Ver(π)). ∗ Then weconsider the linearization z = d ds φ t(p + sv) ∣ s=0. This is a vector at the point φ t (p)which is vertical, since X H is projectable and thus π 10 ◦ φ t = ϕ t ◦ π 10 . Therefore,there exists a vector P t (p, v) ∈ Ver(π) ϕs(m) whose vertical lifting to the pointφ t (p) is the above vector z, i.e. z = ξ V (φ t (p), P t (p, v)). Then we have the mapΨ t (p, v) = (φ t (p), P t (p, v)), which is the parallel transport map along the flow ofX H (and hence along horizontal curves).In other words, if we consider the inverse κ: Ver(π 10 ) → π10(Ver(π)) ∗ of thevertical isomorphism ξ V : π10(Ver(π)) ∗ → Ver(π 10 ) ⊂ T (J 1 π), then the map Ψ t isdefined byΨ t = κ ◦ T φ t ◦ ξ V .Therefore, it follows that for every vectorfield Y ∈ X(E) vertical over R, we haveD X H Y = d dt (Ψ t ◦ Y ◦ φ t ) ∣ ∣t=0,which is equivalent to the relation D X H Y = κ([X H , Y V ]).For parallel transport along vertical curves we can prescribe a complete parallelismrule as follows. Let γ : [a, b] → J 1 π be a curve in the fibre π10 −1 (m), i.e.π 10 (γ(t)) = m, and let p i = γ(a) and p f = γ(b) the endpoints of the curve. We takean element of our bundle z ∈ π10(Ver(π)) ∗ at the initial point p i , that is z = (p i , v)for some v ∈ Ver(π) m . Then the parallel transport of z from p i to p f along thecurve γ is Pp γ i,p f(p i , v) = (p f , v). It is clear that parallel transport is independentof the curve γ that joins the point p i and p f as long as this curve is vertical, and itis in this sense that we speak about complete parallelism.5. Parallel distributions are integrableThe relation between parallel transport and Lie transport that we have seen inthe last section, suggests that there must be some relation between the propertiesfor a distribution of being parallel and being invariant under Lie transport by theflows in the distribution, which is known to be equivalent to the integrability of thedistribution. In the case we are considering (the case of the sode-connection) withthe help of the torsion-free condition we can see that parallel distributions alongπ 10 are integrable. The precise meaning of this terminology is as follows.

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

PARALLEL TRANSPORT AND DECOUPLING 9[9] Martínez E and Cariñena JF Geometric characterization of linearizable second-orderdifferential equations Math. Proc. Camb. Phil. Soc. 119 (1996) 373–381[10] Mestdag T and Sarlet W The Berwald-type connection associated to time-dependentsecond-order differential equations Houston J. Math. 27 (2001), no. 4, 763–797[11] Sarlet W, Crampin M and Martínez E The integrability conditions in the inverse problemof the calculus of variations for second-order ordinary differential equations Acta Appl. Math.54 (1998), no. 3, 233–273Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009 Zaragoza,Spain