On Special Types of Nonholonomic 3-jets

We deduce a classification of all special types of nonholonomic 3-jets. In the introductory part, we summarize the basic properties of nonholonomic r-jets. Generally speaking, a very attractive phenomenon of the problem of classifying the special types of nonholonomic 3-jets is that its solution is heavily based on the Weil algebra technique, even though no algebras appear in the formulation of the problem. We start with summarizing some properties of classical r-jets from the viewpoint used in the present paper and then we mention the basic properties of nonholonomic r-jets, [2]. The second part of Section 1 is devoted to the categorial approach to the concept of special type of nonholonomic r-jets from [7]. In Section 2 we describe the nonholonomic 3-jets in detail. Section 3 contains our previous classification results concerning nonholonomic 2-jets, [5], semiholonomic 3-jets, [3], and two lemmas on the invariant homomorphisms of the related Weil algebras. Section 4 is devoted to the fundamental algebraic properties of the Weil algebra D̃m corresponding to the nonholonomic 3-jets. In Section 5 we classify those nonholonomic 3-jets that are not one-semiholonomic. The classification list is completed in the last section. All manifolds and maps are assumed to be infinitely differentiable. Unless otherwise specified, we use the terminology and notation from [8]. 1. Nonholonomic r-jets. The classical, or holonomic, r-jets X = j xφ of smooth maps φ : M → N form a fibered manifold J(M,N)→M ×N with respect to the source and target projections αX = x ∈M and βX = φ(x) ∈ N . All r-jets form a category J over pointed manifolds (M,x): if X ∈ J x(M,N)y and Z = j yψ ∈ J y (N,Q)z, then Z ◦X = j x(ψ ◦ φ) ∈ J x(M,Q)z. We write Lm,n = J 0 (R,R)0. Then L = ⋃