Project/Area Number  09640126 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  Nippon Institute of Technology 
Principal Investigator 
HASHIMOTO Hideya Nippon Institute of Technology, Associated Professor, 工学部, 助教授 (60218419)

CoInvestigator(Kenkyūbuntansha) 
KODA Takashi Toyama University, Associated Professor, 理学部, 助教授 (40215273)
MASHIMO Katsuya Tokyo University of Agriculture and technology, Associated Professor, 工学部, 助教授 (50157187)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1997 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  octonions / the Lie group G_2 of automorphisms of the octonions O / the 6dimensional unit sphere S^6=G_2 / SU (3) / grassmann subbundles / Jholomorphic curves / CRsubmanifolds / totally real submanifolds / 例外型単純Lie群G2 / グラスマン束 / Jholonorphic curve / totally real submfd / CR部分多様体 / 6次元球面 / ケーリー代数 / 例題型14次元単純Lie群G_2 / 概エルミート構造 / 例外リー群G_2 / J正則曲線 / CR部分多様体 / リーマン3対称空間 
Research Abstract 
Let S^6 be the 6dimensional unit sphere centered at the origin in a 7dimensional Eucliclean spacc. We identified 7dimensional Euclidean space with purely imaginary octonions IrnO (or Cayley algebra). Taking account of algebraic properties of octonions we can define the homogeneous almost Herinitian structure on S^6 We denote by G_2 the Lie group of automorphisms of the octonions 0. Then we have S^6  G_2/SU(3). This almost complex structure satisfy the nearly Kah1er condition ((*xJ)X=0) where * is the LeviCivita connection of S^6, and X is any vector field of S^6 First, we gave the representaion of homogeneous grassmann subbundles corresponding to the invariant submanifolds (Jholomorphic curves, totally real and CRsubmani[olds) of S^6=( S^6, J,<, >). Next, we obtained some constructions of invariant submanifolds of S^6=(S^6, J, < , >). (1). We gave many examples of 3dimensional homogeneous CRsubmanifolds or S^6 and extension cC the 3dimensional CRsubmanifold which is obtained by K.Sekigawa. (2). We obtained some rigidity theorem of 3dimensional CRsubmanifolds up to the action or G2 and determine G2 invariants. (3). We construct 3dimensional totally real submanifolds and 3dimensional CRsubmanifolcls as a tube of the first or second normal bundle of Jholomorphic curves. (4). We give some examples of 4dimensional CRsubmanifolds and studied the obstructions of the existence of such submanifolds.
