Differential geometric study of surfaces with singularities

我给出了虎唇、虎嘴、虎蝶、奇异前沿脐带的标准。进一步给出了蒙德奇异性和高维奇异性的判据。作为它们的应用，我澄清了奇点与曲面微分几何性质之间的关系。此外，我给出了前线的内在公式。利用这个，我给出了关于奇点数的公式。

I gave criteria for cuspidal lips, cuspidal beaks, cuspidal butterfly, umbilics of singularities fronts. Furthermore, I gave criteria for Mond singularity and higher dimensional singularities of fronts.As applications of them, I clarified relationships between singularities and differential geometric properties of surfaces. Furthermore, I gave an intrinsic formulation of fronts.Using this, I gave formulas about numbers of singularities.