ON SPRAYS AND HIGHER ORDER CONNECTIONS

Abstract

The purpose of this note is to formulate the theory of sprays in terms of semiholonomic jetsl (thus we give new proofs of some results obtained by AmbrosePalais-Singer2) and to extend it to higher order connections. Higher order connections have been introduced by C. Ehresmannlb but our definition is different and less general. Details will be given in a paper to appear later. Notations and definitions: Let V, and Vm be paracompact differentiable manifolds (by differentiable we always mean C) of dimensions n and m; for any map f: U c V, Vm, f* will be the prolongation to tangent vectors. Let &(V,,F,G, SC) be a fiber bundle over V, with standard fiber F, structural group G, principal bundle S; if s e G, s will also denote the right translation: h -* hs of SC onto itself; T(JC)/G will denote the vector bundle obtained from the tangent bundle T(XC) by identifying vectors which are transforms of each other under right translations of G. Let p be the map: JC X SC -? SC-1 defined by cp(h,h') = h'h-1; an infinitesimal displacementlc x of the fiber F^(X e V,) is a vector (*p(0,Xh) where Oh is the zero vector and Xh any vector which is also tangent to JC at h e JCx; the origin of 6, in the groupoid SSCis the identity map of Fx; as (*(0h,Xh) = <(P(0hs s*Xhs), the space of all infinitesimal displacements of Fx can be identified with T(SXx)/G. Let JQ(Vm,Vn) be the bundle of all q-jets of V, into Vm, a: J1(Vm,Vn) -V, be the projection sending each jet onto its source; if a is a local lifting of V, into Jl(Vm,Vn) (ao is the identity map of U c Vn), the jet jl owill be called nonholonomic jet of order 2 of V, into Vm; let J2(Vm,Vn) be the space of all these jets. By induction on q, we define J(Vm, Vn). If, for any k, the lifting o of Vn into Jk(Vm,Vn) satisfies: jx(j-'-l.) = o(x) where jk-l is the projection: Jk(Vm,Vn) -. Jk-Cl(Vm, Vn), we get the semi-holonomic jets; let JY(Vm,Vn) be the set of all semi-holonomic q-jets; we have: Jq(Vm, Vn) c Jq(Vm, V). The space Iqf(Vn) (or more briefly if) of semi-holonomic frames of order q is a fiber bundle which is associated to Hq(Vn) (space of holonomic frames); moreover H1 is a principal bundle, with structural group Ln (group of all invertible semi-holonomic q-jets of R into R, with source and target 0). Every y e Lq can be written: ic = E akxj + .. . + l/q! al. ..j .XZ . .. . xjq where the matrix a) e Lj (= Ln = full linear group); y e Ln if and only if the aj,... j are symmetrical. By symmetrization we get a projection Pq of -Iq onto Hq. Let Mq be the kernel of the homomorphism jq-: Lq -, L-~ ; M is an abelian group and its Lie algebra can be identified with R ? (Rn*). We have: -M,n = Ml,n = Ln,n = space of all n X n matrices. We shall write: H? = Vn. Let Sq(Vn) (or more briefly Sq) be the vector space of all q-jets (with source x) of the local liftings of Vn into the tangent bundle T(Vn) and SQ = Ux , Vn S; using semi-holonomic jets, we define Sx and Sq; we have SO = Tx (tangent space