Structural stability theorems for integrable differential forms on 3-manifolds

Abstract

81. STATEMENT OF RESULTS LET A4 BE a compact, connected, orientable C” Riemannian manifold of dimension n and 9’(M) the space of all integrable l-forms of class C’ endowed with uniform C’ topology., i.e. o E a’(M) if and only if w is C’ and o A do = 0. A singularity of w E 9’(M) is a point in M where w vanishes. Denote the union of all singularities of o by Sing(o). By Frobenius Theorem w defines a regular foliation of codimension one on M-Sing(w). A topological equivalence between forms w, n E 9’(M) is a homeomorphism h of M sending leaves of o/M-Sing(o) onto leaves of n/M-Sing(T) and Sing(w) onto Sing(n). The form o E 6’(M) is called C-structurally stable, s 5 r, if there is a neighborhood N(w) in 6”(M) such that any v E N(w) is topologically equivalent to w. The main problem concerning the stability of integrable forms is to characterize the elements of 9’(M) which are C-structurally stable. The first one to deal_ with this problem from the local point of view was I. Kupka[6] and more recently several people, [3, 131. Roughly there are two types of singularities, those x E M for which O, = 0, dw,# 0 and those for which o, and dw, vanish simultaneously. Let x0 E M be a singularity of u such that do,o # 0. The form do induces near x0 a regular foliation 9(o) defined by the vector fields X with ix(do) = 0. This foliation has codimension two, it is tangent to the leaves of w and Sing(w) near x0 is a union of leaves .of 9(o). It follows that there is a system of coordinates (x,, . . . x,) around x0 such that w = a,(~,, x2) dx, + az(x,, xz) dxz.