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

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,