We prove that the level sets of a real C 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 anal...

Morse theory was originally due to Marston Morse [5]. It gives us a method to study the topology of a manifold using the information of the critical points of a Morse function defined on the manifold. Based on the same idea, Morse homology was introduced by Thom, Smale, Milnor, and Witten in various forms. This paper is a survey of some work in this direction. The first part of the paper focuse...

We prove that the level sets of a real C function of two variables near a non-degenerate critical point are of class C 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 also analyse the cases coming from degenerate critical points, corresponding to cusps of G...

Submitted for the MAR11 Meeting of The American Physical Society Dirac Point Degenerate with Massive Bands at a Topological Quantum Critical Point SWAPNONIL BANERJEE, UC Davis, WARREN PICKETT, JUSTIN SMITH, VICTOR PARDO, RAJIV SINGH TEAM — In the band structure of the Skutterudite, as the Sb sublattice in the unit cell is moved slightly retaining the crystal symmetry, the small gap at the Fermi...

Submitted for the CAL10 Meeting of The American Physical Society Dirac Point Degenerate with Massive Bands at a Topological Quantum Critical Point1 SWAPNONIL BANERJEE, WARREN PICKETT, JUSTIN SMITH, VICTOR PARDO — In the band structure of the Skutterudite, as the Sb sublattice in the unit cell is moved slightly retaining the crystal symmetry, the small gap at the Fermi energy closes due to a ban...

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of...