Classical differential geometry from the modern viewpoint and its applications

在这项研究中，我们计划从现代的观点，特别是可积系统和奇点的理论，重构经典微分几何理论，给出经典微分几何理论的一个新的发展。我们的主要结果如下：1。(1)在经典微分几何的核心理论之一仿射微分几何中，主要研究了仿射超球的几何及其表示公式，并给出了仿射超球与全纯统计流形几何的关系及其中心映射的若干性质。我们还研究了仿射平面曲线和向心仿射平面曲线的离散化，并根据离散孤子方程描述了它们的时间演化；(2)对四维欧几里得空间中共形平面超曲面的经典实例进行了刻画，并构造了新的实例；(3)对于复杂空间形式的实超曲面，引入了一个新的几何不变量，并利用该不变量对Hopf实超曲面进行了分类。研究了具有奇异点的曲面的几何性质，得到了以下结果：(1)构造了三维双曲空间中一类具有奇异点的平面——平面锋理论。特别地，我们定义了平面前沿的（弱）完备性，并证明了它们的全局性质；(2)研究了三维Minkowski空间中极大曲面奇异性的性质，构造了允许某种奇异性的类空间极大曲面理论。研究了曲面的变换，证明了由莫比乌斯几何中的球同余给出的变换可以由欧几里德空间中的复化线同余得到。我们还研究了三维齐次空间中的双调和曲线，并确定了齐次空间不可约和可约时的双调和曲线。

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.