On restricted analytic gradients on analytic isolated surface singularities

Let (X,0)(X,0) be a real analytic isolated surface singularity at the origin 0 of RnRn and let g be a real analytic Riemannian metric at 0∈Rn0∈Rn. Given a real analytic function f0:(Rn,0)→(R,0)f0:(Rn,0)→(R,0) singular at 0, we prove that the gradient trajectories for the metric g|X∖0g|X∖0 of the restriction (f0|X)(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∖0X∖0 where the restricted gradient does not vanish, there is always a trajectory accumulating at 0 and admitting a formal asymptotic expansion at 0.