Level Sets of Functions and Symmetry Sets of Surface Sections

摘要

We prove that the level sets of a real C~s function of two variables near a non-degenerate critical point are of class C~([s/2]) and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also analyse one of the cases coming from a degenerate critical point, corresponding to an elliptic cusp of Gauss on a surface, where the differentiability is reduced to C~([s/4]). However in all our applications we assume C~∞ smoothness.