On restricted analytic gradients on analytic isolated surface singularities

Abstract

Let (X,0) be a real analytic isolated surface singularity at the origin 0 of Rn and let g be a real analytic Riemannian metric at 0∈Rn. Given a real analytic function f0:(Rn,0)→(R,0) singular at 0, weprove that the gradient trajectories for the metric g|X\0 of the restriction (f0|X) escaping from or ending up at 0 do not oscillate. Such a trajectory is thus a sub-pfaffian set. Moreover, in each connected component of X\0 where the restricted gradient does not vanish, there is always a trajectory accumulating at 0 and admitting a formal asymptotic expansion at 0.