The constructions of general connections on second jet prolongation

Abstract

we denote the set of all r-th order linear connections on M .A classical linear connection on M is a first order linear connectionwe denote the set of all torsion free classical linear connections on M .Let FM denote the category of fibred manifolds and their fibred maps and let FM m,n ⊂ FM be the (sub)category of fibred manifolds with mdimensional bases and n-dimensional fibres and their local fibred diffeomorphisms. Let Mf m denote the category of m-dimensional manifolds and their local diffeomorphisms. Let F : FM m,n → FM be a bundle functor on FM m,n of order r in the sense of [4]. Let Γ :In other words, the value Fη(u) at every u ∈ F y Y, y ∈ Y depends only on j r y η. Therefore, we have the corresponding flow morphismF :is called F -prolongation of Γ with respect to Λ and was discovered by I. Kolář [5]. Let ∇ be a torsion free classical linear connection on M . For every x ∈ M , the connection ∇ determines the exponential map exp, which is diffeomorphism of some neighbourhood of the zero vector at x onto some neighbourhood of x. Every vector v ∈ T x M can be extended to a vector fieldṽ on a vector space T x M byṽ(w) = ∂ ∂t |t=0[w+tv]. Then we can construct an r-th order linear connection E r (∇) :This connection is called an exponential extension of ∇ and was presented by W. Mikulski in [9]. Another equivalent definition (for corresponding principal connections in the r-frame bundles)The constructions of general connections on second jet prolongation 69 of the exponential extension was independently introduced by I. Kolář in [6]. Hence given a general connection Γ on Y → M and a torsion free classical linear connection ∇ on M , we have the general connectionThe canonical character of construction of this connection can be described by means of the concept of natural operators. The general concept of natural operators can be found in [4]. In particular, we have the following definitions.commutes. We say that the operator D Y is regular if it transforms smoothly parametrized families of connections into smoothly parametrized ones.commutes, too. The regularity means that every A M transforms smoothly parametrized families of connections into smoothly parametrized ones.Thus the construction F(Γ, Λ) can be considered as the FM m,n -natural operator F :J 1 (F → B). Similarly, the correspondence E r : Q τ Q r is the Mf m -natural operator. In [4], the authors described all FM m,n -natural operators D :In this paper we determine all FM m,n -natural operators D :. We assume that all manifolds and maps are smooth, i.e. of class C ∞ . Quasifor some (uniquely determined) real numbers a From the proof of Proposition 2.2 from [8] it follows that (B × H)• ψ is a (Γ, ∇, y 0 , 3)-quasi-normal fibred coordinate system for any B ∈ GL(m) and any diffeomorphism H : R n → R n preserving 0. In other words, the FM m,n -maps of the form B × H for B ∈ GL(m) and diffeomorphisms H : R n → R n preserving 0 ∈ R n transform quasi-normal fibred coordinate systems into quasi-normal ones.From now on we will usually work in (Γ, ∇, y 0 , ...

The constructions of general connections on the fibred product of q copies of the first jet prolongation

“…where P and J 1 are natural operators constructed in the monograph [4]. In [8], we generalized this result to the case F = J 2 . In other words, we classified all FM m,n -natural operators D :…”