Group action and Grassmann Geometry
Project/Area Number  13640063 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Tokyo University of Agriculture and Technology 
Principal Investigator 
MASHIMO Katsuya Tokyo University of Agriculture and Technology, Faculty of Technology, Professor, 工学部, 教授 (50157187)

CoInvestigator(Kenkyūbuntansha) 
TOJO Koji Chiba Institute of Technology, Faculty of Technology, Associate Professor, 工学部, 助教授 (30296313)
TASAKI Hiroyuki University of Tsukuba, Intsitute of Mathematics, Associate Professor, 数学系, 助教授 (30179684)
HASHIMOTO Hideya Meijo University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60218419)
古田 高士 富山大学, 理学部, 助教授 (40215273)

Project Period (FY) 
2001 – 2003

Project Status 
Completed(Fiscal Year 2003)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 2003 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2002 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 2001 : ¥1,200,000 (Direct Cost : ¥1,200,000)

Keywords  6 dimensional sphere / G_2 / Grassmann geometry / CR submanifolds / J holomorphic curve / tube / CR submanifold / Chen's inequality / Spin(7) / austere部分多様体 / ポアンカレの公式 / 全実部分多用体 / CR部分多用体 
Research Abstract 
We studied submanifolds of S^6 from the viewpoint of Grassmann geometry. The compact simple Lie group of type G_2 acts on S^6. Extend the action to the action on the Grassmann bundle G^p (TS^6) of all pdimensional subspaces of a tangent space of S^6 and decompose G^p (TS^6). Take a single orbit Σ of the action. We say that a pdimensional submanifold N is a Σsubmanifokd if and only if the tangent space of N is contained in Σ. As a special case of Σsubmanifold, we have the class of totally real submanifolds and CRsubmanifolds. (1)Submanifolds which are obtained as a tube over another submanifolds are studied by many nathematicians. We classified all 3dimensional submanifolds which are obtained as tubes in the direction of first or second normal space over a Jholomorphic curve. Concerning the problem, Hashimoto studied a method of construction of Jholomorphic curves from 2tori to S^6 by a joint work with Taniguchi and Udagawa. (2)We have a general inequality, proved by B.Y.Chen in 1991, for submanifolds of space forms. We studied minimal CR submanifolds of S^6 which attains the equality in Chen's inequality. Similar problem were studied by Dillen and Vrancken for 3dimensional totally real submanidolds of S^6.

Report
(4results)
Research Products
(9results)