Classical differential geometry from the modern viewpoint and its applications

在这项研究中，我们计划通过从现代的观点重组经典微分几何理论，特别是可积系统理论和奇点理论，来给出一个现在的发展。1.(1)在经典微分几何的核心理论之一仿射微分几何中，我们主要研究了仿射超球面的几何及其表示式，并证明了它与全纯统计流形的几何以及中心映射的几个性质之间的关系。我们还研究了仿射或中心仿射平面曲线的离散化，并用离散孤子方程描述了它们的时间演化过程；(2)刻画了4维欧氏空间中共形平坦超曲面的经典例子，构造了新的例子；(3)对于复空间形式中的实超曲面，我们引入了一个新的几何不变量，并利用不变量对Hopf实超曲面进行了分类。2.研究了奇点曲面的几何性质，得到了以下结果：(1)我们建立了平坦面理论，即三维双曲空间中具有某种奇点的平坦曲面。研究了三维Minkowski空间中极大曲面奇点的性质，构造了允许某种奇点的类空极大曲面；3.研究了曲面的变换，证明了Moebius几何中球同余给出的变换是由欧氏空间中的复化直线同余得到的。我们还研究了三维齐次空间中的双调和曲线，并确定了当齐次空间不可约和可约时的重调和曲线。

In this research, we planned to give a now development of the theories of classical differential geometry by restructuring them from the modern viewpoint, particularly, of the theories of integrable systems and of singularities. Our main results are the following :1.(1)In affine differential geometry, one of the core theories of classical differential geometry, we mainly studied the geometry of affine hyperspheres and their representation formulae, and showed a relationship with the geometry of holomorphic statistical manifolds and the several properties of the center maps. We also studied the discretization of affine or centroaffine plane curves and gave a description of their time-evolution following discrete soliton equations ; (2)we characterized the classical examples of conformally flat hypersurfaces in 4-dimensional Euclidean space and constructed new examples ; (3)for real hypersurfaces in complex space forms, we introduced a new geometric invariant and classified Hopf real hypersurfaces using the invariant.2.We studied the geometric properties of surfaces with singularities and obtained the following results : (1)We constructed the theory of flat fronts, the flat surfaces with singularities of a certain kind in 3-dimensional hyperbolic space. In particular, we defined (weak) completeness of flat fronts and showed their global properties ; (2)investigating the properties of the singularities of maximal surfaces in 3-dimensional Minkowski space, we constructed the theory of maxfaces, the spacelike maximal surfaces allowing singularities of a certain kind.3.We studied transformations of surfaces and showed that the transformations given by the sphere congruences in Moebius geometry are obtained by the complexified line congruences in Euclidean space. We also investigated biharmonic curves in 3-dimensional homogeneous spaces and determined such curves when the homogeneous spaces are irreducible and reductive.