ON THE HOLONOMIZATION OF SEMIHOLONOMIC JETS

Abstract

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