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

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 ]).