Exceptional Lie Groups and Grassmann Geomtry

例外李群和格拉斯曼几何

• 批准号: 09640126
• 负责人: HASHIMOTO Hideya
• 金额: $ 1.92万
• 依托单位: Nippon Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640126/
• 关键词: octonions; the Lie group G_2 of automorphisms of the octonions O; the 6-dimensional unit sphere S^6=G_2; SU (3); grassmann subbundles; J-holomorphic curves; CR-submanifolds; totally real submanifolds; 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

Let S^6 be the 6-dimensional unit sphere centered at the origin in a 7-dimensional Eu-cliclean spacc. We identified 7-dimensional Euclidean space with purely imaginary octo-nions 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 Levi-Civita connection of S^6, and X is any vector field of S^6First, we gave the representaion of homogeneous grassmann subbundles corresponding to the invariant submanifolds (J-holomorphic curves, totally real and CR-submani[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 3-dimensional homogeneous CR-submanifolds or S^6 and extension cC the 3-dimensional CR-submanifold which is obtained by K.Sekigawa.(2). We obtained some rigidity theorem of 3-dimensional CR-submanifolds up to the action or G2 and determine G2 invariants.(3). We construct 3-dimensional totally real submanifolds and 3-dimensional CR-submanifolcls as a tube of the first or second normal bundle of J-holomorphic curves.(4). We give some examples of 4-dimensional CRsubmanifolds and studied the obstructions of the existence of such submanifolds.

令 S^6 为 7 维 Eu-cliclean 空间中以原点为中心的 6 维单位球面。我们用纯虚八元 IrnO（或凯莱代数）确定了 7 维欧几里得空间。考虑到八元数的代数性质，我们可以定义 S^6 上的齐次几乎赫里尼特结构。我们用 G_2 表示八元数 0 的自同构李群。然后我们有 S^6 - G_2/SU(3)。这个几乎复杂的结构满足近 Kah1er 条件 ((*xJ)X=0)，其中 * 是 S^6 的 Levi-Civita 连接，X 是任意 S^6的向量场首先给出了S^6=( S^6, J,<, >)的不变子流形(J-全纯曲线、全实曲线和CR-submani[olds)对应的齐次格拉斯曼子丛的表示。接下来，我们得到了S^6=(S^6, J, < , >).(1).我们给出了许多 3 维齐次 CR 子流形或 S^6 的例子，以及由 K.Sekikawa (2) 获得的 3 维 CR 子流形的扩展 cC。我们得到了3维CR-子流形的一些刚性定理直到G2的作用并确定了G2不变量。(3)。我们构造了3维全实子流形和3维CR-子流形作为J-全纯曲线第一或第二法束的管子。(4)．我们给出了一些4维CR子流形的例子，并研究了这种子流形存在的障碍。

项目成果

T.Koda,H.Hashimoto: "Grassmann Geometry of 6-dimensional sphere" Topics in complex analysis,Dift Geom and Math.Physis. 136-142 (1997)

T.Koda，H.Hashimoto：“六维球体的格拉斯曼几何”复分析、Dift Geom 和 Math.Physis 主题。

H.Hashimoto and S.Funabashi.: "Hypersurfaces in a 6-dimensional sphere." J.Korean Math.Soc.23-42 (1997)

H.Hashimoto 和 S.Funabashi.：“六维球体中的超曲面。”

T.Koda and Y.Watanabe.: "Homogeneous almost contact Riemannian manifolds and its infinitesimal model." Bollenttino U.M.I.,. 11-B. 11-24 (1997)

T.Koda 和 Y.Watanabe.：“齐次几乎接触黎曼流形及其无穷小模型。”

間下克哉: "On branching theorem of the pair (G_2,Su(3))" Nihonkai Math.J. 8. 101-107 (1997)

Katsuya Mashita：“论对的分支定理 (G_2,Su(3))” Nihonkai Math.J. 8. 101-107 (1997)

H.Hashimoto,S.Funabashi: "Hypersurfaces in a 6-dimensional sphere" J.korean.Math.Soc. 34. 23-42 (1997)

H.Hashimoto，S.Funabashi：“六维球体中的超曲面”J.korean.Math.Soc。