Geometry of Gauss Mapping

首先研究了球面S^n上具有退化高斯映射的子流形M。M到实格拉斯曼的高斯映射当且仅当M在S^n中是完全测地线时是常数，所以高斯映射的秩测量M的形状接近完全测地线的程度。另一方面，每片叶子的叶理由核给出的高斯映射的微分。关键问题是，对于由大球体叶状结构的S^n中的子流形M，找出高斯映射沿每个叶状结构都是常数的条件。本文利用实格拉斯曼尼亚上的正则球束，给出了构造S^n中由大球叶化的子流形的一般方法。此外，对于复二次曲面上的复子流形上的圆束或四元数对称空间上的扭转空间，我们证明了沿子流形上球束的每一根纤维的高斯映射是常数的，并且它们的扭转法锥在复欧几里德空间中是特殊的拉格朗日。

First we investigated submanifolds M with degenerate Gauss mapping in spheres S^n. The Gauss map of M to real Grassmannian is constant if and only if M is totally geodesic in S^n, so the rank of the Gauss map measures the degree of how shape of M is near to the totally geodesic one. On the other hand, each leaf of the foliation given by the kernel of the differential of the Gauss map. So essential problem is that for a submanifold M in S^n foliated by great spheres, find the condition of which along each leaf the Gauss map is constant. In this research, we give general method to construct submanifolds foliated by great spheres in S^n by using the canonical sphere bundle over real Grassmanniam. Moreover, for either a circle bundle over complex submanifolds in complex quadrics or the twistor space over quaternionic symmetric spaces, we showed that along each fiber of the sphere bundles over submanifolds, the Gauss map is constant, and their twisted normal cones are special Lagrangian in complex Euclidean space.