Plotting region with negative Gaussian curvature?

There is a ResourceFunction that can compute Gaussian curvature which allows you to incorporate the comment by Ulrich Neumann

Clear[surf, curv]
surf[u_, v_] = ResourceFunction["Cyclide"][.3, 1.5, {u, v}];
curv[u_, v_] = 
  ResourceFunction["GaussianCurvature"][#, {u, v}] &@surf[u, v];
ParametricPlot3D[
 surf[u, v], {u, -[Pi], [Pi]}, {v, -2 [Pi], 2 [Pi]}, 
 RegionFunction -> Function[{u, v}, curv[u, v] < 0]]