Gradient Vector Fields Do Not Generate Twister Dynamics

Thomʹs Gradient Conjecture states that a solution γ of an analytic gradient vector field X approaching to a singularity P of X has a tangent at P. A stronger version asserts that γ does not meet an analytic hypersurface an infinite number of times (it is non-oscillating). We prove, in dimension 3, that if γ is “infinitely near” an analytic curve Γ not composed of singularities of X, then γ is non-oscillating and, moreover, it does not spiral around Γ in a precise sense