Non oscillating solutions of analytic gradient vector fields

Abstract

Let γ be an integral solution of an analytic real vector field ξ defined in a neighbordhood of 0∈ℝ 3 . Suppose that γ has a single limit point, ω(γ)={0}. We say that γ is non oscillating if, for any analytic surface H, either γ is contained in H or γ cuts H only finitely many times. In this paper we give a sufficient condition for γ 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 ξ=∇ g f of an analytic function f of order 2 at 0∈ℝ 3 , where g is an analytic riemannian metric.