Two significant directions in the development of jet calculus are showed. First, jets are generalized to so-called quasijets. Second, jets of foliated and multifoliated manifold morphisms are presented. Although the paper has mainly a survey character, it also includes new results: jets modulo multifoliations are introduced and their relation to (R,S,Q)-jets is demonstrated.
Using a general connection Γ on a fibred manifold p:Y → M and a torsion free classical linear connection ∇ on M, we distinguish some “special” fibred coordinate systems on Y, and then we construct a general connection on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.
We deduce a classification of all special types of nonholonomic -jets. In the introductory part, we summarize the basic properties of nonholonomic -jets.
First we summarize some properties of the nonholonomic -jets from the functorial point of view. In particular, we describe the basic properties of our original concept of nonholonomic -jet category. Then we deduce certain properties of the Weil algebras associated with nonholonomic -jets. Next we describe an algorithm for finding the nonholonomic -jet categories. Finally we classify all special types of semiholonomic -jets.
Let Mⁿ (n ≥ 3) be an n-dimensional complete hypersurface in a real space form N(c) (c ≥ 0). We prove that if the sectional curvature of M satisfies the following pinching condition: , where δ = 1/5 for n ≥ 4 and δ = 1/4 for n = 3, then there are no stable currents (or stable varifolds) in M. This is a positive answer to the well-known conjecture of Lawson and Simons.
Let and be fiber product preserving bundle functors on the category of fibred manifolds with -dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism to be a -invariant algebra homomorphism with . The main result is that there exists an -natural transformation depending on a classical linear connection on the base of if and only if there exists a quasi-morphism . As applications, we study existence problems of symmetrization (holonomization)...
Automorphisms of the family of all Sturm-Liouville equations are investigated. The classical Darboux transformation arises as a particular case of a general result.