Non oscillating solutions of analytic gradient vector fields

Abstract

Let \gamma be an integral solution of an analytic real vector field defined in a neighbordhood of 0\in R3. Suppose that \gamma has a single limit point at 0. We say that \gamma is non oscillating if, for any analytic surface H, either \gamma is contained in H or \gamma cuts H only finitely many times. In this paper we give a sufficient condition for \gamma to be non oscillating. It is established in terms of the existence of “generalized iterated tangents”, i.e. the existence of a single limit point for any transform property for the solutions of a gradient vector field 𝛻g f of an analytic function f of order 2 at 0, where g is an analytic riemannian metric.