A criterion for the nonexistence of closed leaves in unorientable planar foliations

The family of all smooth foliations on an open set is naturally parameterized by all smooth maps , in the sense that the values determine the tangent line to the leaf of at p Ω. If is further assumed to be orientable, a smooth global branch Y of the square root of X can be chosen. In this case, one has the classical Lyapunov criterion: if there is a real-valued u C1(Ω) such that is nowhere zero, then has no closed leaves (the vector field Y has no periodic orbits). In this paper we introduce an analytic criterion for the nonexistence of closed leaves, similar in spirit to that of Lyapunov, but which allows for to be unorientable as well. The possible lack of orientability makes the replacement for the first order differential operator Y considerably more involved. In fact, one has to work with a second order linear hyperbolic differential operator whose coefficients carry information about the curvature of the leaves of . It is shown that if is given by X:Ω→S1 in the manner described above, and there exists a real-valued u C2(Ω) such that is nowhere zero, then has no closed leaves. We apply the new criterion when X is holomorphic, providing also an example that shows the need for the first order term in the definition of .