Rigid paths of generic 2-distributions with degenerate points on 3-manifolds

Abstract

We study rigid paths of generic 2-distributions with degenerate points on 3-manifolds. A complete description of such paths is obtained. For the proof, we construct separating surfaces of paths admissible for distributions. 0. Introduction. The purpose of this paper is to give a complete explicit description of rigid paths of generic 2-distributions, with degenerate points, on 3-manifolds. By a 2-distribution with degenerate points on a 3manifold, we mean what is represented by either of the following objects: (a) a Pfaffian equation {ω = 0}, where ω is a smooth differential 1-form, (b) a module 〈X,Y 〉 of vector fields over the ring of smooth functions, which is generated by smooth vector fields X and Y . In the following, we call such 2-distributions just “2-distributions” for convenience. For a usual plane field {Dp ⊂ TpM}p∈M on a 3-manifold M , a tangent plane Dp has the constant dimension dimDp = 2 at any point p ∈ M . We note that a 2-distribution E, considered in this paper, may have a point p ∈ M where dimEp is 0, 1 or 3. Such 2-distributions on 3-manifolds were studied by B. Jakubczyk and M. Ya. Zhitomirskĭi in [JZh]. They gave a complete description of singularities of such distributions and a list of local normal forms in each case, (a) and (b). Let M be a smooth connected manifold and E a 2-distribution on M . A path γ : [α, β] → M is called admissible for E if it is tangent to E at any point p ∈ Im γ. Let a, b ∈ M be given two points. We denote the space of all admissible paths joining a to b by ΩE(a, b) := {γ : [0, 1] → M admissible | γ(0) = a, γ(1) = b}. A path γ ∈ ΩE(a, b) is called rigid if any path in ΩE(a, b) C-close enough to γ has the same image as γ. This notion was introduced in [BH]. Rigid paths of generic 2-distributions without degenerate points on 3-manifolds 2000 Mathematics Subject Classification: 58A17, 58A30, 53C15.