Włodzimierz M. Mikulski

On the fiber product preserving bundle functors

Abstract We present a complete description of all fiber product preserving bundle functors on the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. This result is based on several general properties of such functors, which are deduced in the first two parts of the paper.

Holonomic extension of connections and symmetrization of jets

Taking into account motivations from the geometrization of mathematical physics, we discuss the symmetrization of higher order jets and extension of connections. First we introduce an extension of a general connection Γ on a fibered manifold Y → M into an r -th order holonomic connection and we classify all second-order holonomic extensions of Γ. Then we study the general problem of symmetrization of jets. We introduce symmetrizations of higher-order semiholonomic and nonholonomic prolongations and we describe all symmetrizations of an r -th order semiholonomic prolongation j ¯ r .

Some natural constructions on vector fields and higher order cotangent bundles

We prove that forn-manifolds (n≥3) the sets of all natural operatorsT→(Tr*,Tq*) andT-TTr*, respectively, are free finitely generatedC∞(Rr)-modules. We construct explicitly the bases of theC∞(Rr)-modules.

Negative answers to some questions about constructions on connections

Let m and n be natural numbers. For an arbitrary bundle functor G on the category TM.m,n of fibred manifolds with m-dimensional bases and n-dimensional fibers and their local fibered diffeomorphisms we prove that there is no -F.Mm,n-natural operator T> transforming general connections F on Y M and classical linear connections V on M into classical linear connections T>(T, V) on GY. Some generalization of this result is also presented. For an arbitrary gauge bundle functor G on the category VBm,n of vector bundles with m-dimensional bases and n-dimensional fibers and their local vector bundle isomorphisms we prove that there is no VBm,n-natural operator D transforming linear general connections r on E —> M into classical linear connections D(T) on GE. For a Lie group G and an arbitrary gauge bundle functor G on the category Vm{G) of principal G-bundles with m-dimensional bases and their local G-bundle isomorphisms we prove that there is no Pm(G)-natural operator T> transforming G-invariant general connections T on P —> M into classical linear connections 2?(r) on GP. These results give negative answers to some interesting questions about constructions on connections. 0. Introduction Let Y —> M be a fibred manifold. A general connection on Y —• M is a section T : Y JY of the first jet prolongation JY -» Y of Y -> M, which can be considered as the corresponding lifting map (denoted by the same letter) T : F x M TM —> TY . If Y = E is a vector bundle, then a general connection T : E —> J E on E is called linear if it is a vector bundle map.

On the contact (k, r)-coelements

For natural numbers n, r and k with n > fc the bundle functor of contact (fc, r)-coelements over n-manifolds is denoted by Kk*. The rigidity theorem for Kk* is proved. If n > k(r + 1) the natural operators T\Mfn TKk* and T*Mfn T*K{* are completely described and the natural affinors on Kk* are classified. The case r = fc = 1 is additionally discussed. 0. Introduction Let n, r and fc be natural numbers. Let n > fc. In [2], C. Ehresmann constructed functorially the fibre bundle KkM = regTkM / Lk over a n-dimensional manifold M of contact (fc, r)elements and obtained the bundle functor Kk : M f n —> TM. from the category M f n of n-dimensional manifolds and their embeddings into the category TM. of fibered manifolds and their fibered maps. In [5], I. Kolar, P.W. Michor and J. Slovak studied the problem how a vector field X on M induces a vector field A(X) on K£M and proved that every natural operator A : T l M f n —» TK£ is a constant multiple of the complete lifting K,k. In [6], 1. Kolar and the author investigated the naturality problem with bundle mappings B : K£M —> K£M and deduced the so called rigidity theorem for KJ. saying that every natural transformation B : Kjl over n-manifolds is the identity one. The authors studied also the naturality problem with affinors (i.e. tensor fields of type (1,1)) C : TKkM TKkM on KkM and derived that for n > fc + 2 every natural affinor C : TKk —> TKk on Kk over n-manifolds is a constant multiple of the identity one. Moreover the authors analysied how a 1-form u; on M can induce a 1-form D(u) on KkM and showed that every natural operator D : T j ^ y T*Kk is a constant multiple of the vertical lifting. 1991 Mathematics Subject Classification: 58A05, 58A20.

Natural base-extending operators of foliations into foliations on the Weil functors

For any Weil algebraA we establish a bijection between the set of all natural base-extending operators of foliations into foliations on the Weil functor ofA-velocities and the set of all pairs (I, J) of ideals inA withI ⊃J.

Lagrangians and Euler morphisms from connections on the frame bundle

We classify all natural operators transforming torsion free classical linear connections ∇ on m‐dimensional manifolds M into r‐th order Lagrangians λ(∇) and Euler morphisms E(∇) on the linear frame bundle P1M. We also briefly write how this classification result can be generalized on higher order frame bundles PkM instead of P1M.

Natural operators lifting projectable-projectable vector fields to product preserving bundle functors on fibered-fibered manifolds

For any product-preserving bundle functor F defined on the category TbA of fibered-fibered manifolds, we determine all natural operators transforming projectableprojectable vector fields on Y 6 Ob(J-^SA) to vector fields on FY. We also determine all natural afRnors on FY and prove a composition property analogous to that concerning Weil bundles. 0. Preliminaries The classical results by Kainz and Michor, [2], Eck, [1] and Luciano, [5] read that the product preserving bundle functors on the category Mf of manifolds are just Weil bundles, [4], Let us remind the result by Kolar, [3]. For a bundle functor F : Mf —> FM, denote by T the flow operator associated to F which is defined by TX = F(Fl?) for any vector field X on M. Further, consider an element c of a Weil algebra A and let L(c)M • TTM TTM denote the natural affinor by Koszul, [3] and [4]. Then we have a natural operator L(C)M°T : TM TTM lifting vector fields on a manifold M to a Weil bundle TM. As for the absolute natural operators T —• TT, i.e. independent on a vector field X, they are of the form AD for a derivation D € Der A. They are defined as follows. The Lie algebra Aut(A) associated to the Lie group of all algebra automorphisms of A is identified with the algebra of derivations Der A of A. For 1991 Mathematics Subject Classification: 58A20.

The Cauchy-Kowalevski theorem applied for counting connections with a prescribed Ricci tensor

How many linear connections are there with a prescribed Ricci tensor? The question is answered in the analytic case by using the Cauchy–Kowalevski theorem.

ON THE HOLONOMIZATION OF SEMIHOLONOMIC JETS

Abstract. We find all FM m -natural operators Atransforming torsionfree classical linear connections ∇on m-manifolds M into base preservingfibred maps A(∇) : J r Y → J r Y for FM m -objects Y with bases M,where J r , J r are the semiholonomic and holonomic jet functors of orderr on the category FM m of fibred manifolds with m-dimensional basesand their fibred maps with embeddings as base maps. 0. IntroductionAll manifolds considered in the paper are assumed to be finite dimensional,without boundaries, Hausdorff, second countable and smooth (of class C ∞ ).Maps between manifolds are assumed to be of class C ∞ .The classical theory of higher order jets was introduced by C. Ehresmann,[2]. For semiholonomic jets, we refer to the paper by P. Libermann, [8]. Higherorder jets are a very powerful tool in differential geometry and in mathematicalphysics. For example, holonomic jets globalize the theory of differential systemsand semiholonomic jets play an important role in the calculus of variations andin the theory of partial differential equations, [11], [12]. The theory of jets andconnections forms the geometrical background for field theories and theoreticalphysics, [7], [9]. Holonomic and semiholonomic prolongation functors J