Geometry of Gauss Mapping

高斯映射的几何

• 批准号: 13640073
• 负责人: KIMURA Makoto
• 金额: $ 2.05万
• 依托单位: Shimane University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640073/
• 关键词: Differential Geometry; Gauss map; Minimal submanifolds; Special Lagrangian; 特殊ラグランジュ性; グラスマン多様体; 特殊ラグランジュ部分多様体; ツイスター空間; 四元数対称空間; 合同性

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.

首先研究球面S^n中具有退化Gauss映射的子流形M。M到真实的Grassmannian的Gauss映射是常数当且仅当M在S^n中是全测地的，因此Gauss映射的秩度量了M的形状接近全测地的程度。另一方面，叶理的每一片叶子由高斯映射的微分核给出。因此，对于S^n中的一个由大球面构成的子流形M，求出高斯映射沿其沿着每一叶为常数的条件。本文利用真实的Grassmanniam上的典型球丛给出了S^n中由大球面构成的子流形的一般构造方法。此外，对于复二次曲面中的复子流形上的圆丛和四元数对称空间上的扭量空间，证明了在复欧氏空间中，沿子流形上球丛的沿着每根纤维，高斯映射是常数，其扭法锥是特殊的拉格朗日函数.

项目成果

U-Hang Ki., M,Kimura., S,Maeda.: "Geometry of holomorphic distributions of real hypersurfaces in a complex projective space"Czec. Math. J.. 51. 197-204 (2001)

U-Hang Ki.，M，Kimura.，S，Maeda.：“复杂射影空间中真实超曲面的全纯分布的几何”捷克语。

T.Adachi, S.Maeda: "Length spectrum of geodesic spheres in a non-flat complex space form"J.Math.Soc.Japan. (to appear).

T.Adachi、S.Maeda：“非平坦复空间形式中测地线球体的长度谱”J.Math.Soc.Japan。

K.Suizu, S.Maeda, T.Adachi: "Characterization of totally geodesic kahler immersions"Hokkaido Math. J.. 31. 629-641 (2002)

K.Suizu、S.Maeda、T.Adachi：“完全测地线卡勒沉浸的表征”北海道数学。

G.Ishikawa, M.Kimura, R.Miyaoka: "Submanifolds with degenerate Gauss mapping in spheres"Adv.Studies in Pure Math.. 37. 115-149 (2002)

G.Ishikawa、M.Kimura、R.Miyaoka：“球体中具有简并高斯映射的子流形”Adv.Studies in Pure Math.. 37. 115-149 (2002)

G.Ishikawa, M.Kimura, R.Miyaoka: "Submanifolds with degenerate Gauss mappings in spheres"Adv. Studies in Pure Math.. 37. 115-149 (2002)

G.Ishikawa、M.Kimura、R.Miyaoka：“球体中具有简并高斯映射的子流形”Adv。