Integrable homogeneous geometric structures and invariants

2015 Fiscal Year Final Research Report

• Project/Area Number: 23540090
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Hiroshima University
• Principal Investigator: AGAOKA Yoshio 広島大学, 理学(系)研究科(研究院), 教授 (50192894)
• Co-Investigator(Kenkyū-buntansha): TAMARU Hiroshi 広島大学, 大学院理学研究科, 教授 (50306982) ; SHIBUYA Kazuhiro 広島大学, 大学院理学研究科, 准教授 (00569832)
• Co-Investigator(Renkei-kenkyūsha): KONNO Hitoshi 東京海洋大学, 海洋工学部, 教授 (00291477)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Keywords: 可積分条件 / 共形構造 / 平坦 / plethysm / ガウス方程式 / 不変式 / 等長埋め込み / コダッチ方程式

Outline of Final Research Achievements

In this research we investigate the problem of the existence or non-existence of integrable geometric structures on homogeneous spaces, and also the problem of the integrability condition on local isometric imbeddings of Riemannian manifolds from the standpoint of algebraic invariant theory. In addition, we also investigate several fundamental problems on representation theory such as a decomposition formula of plethysms etc.
We obtained several results on these problems. For example, we construct a geometric invariant associated with left invariant flat conformal structures on Lie groups, and also give a necessary and sufficient condition for 3-dimensional Riemannian manifolds that are locally isometrically imbedded into the 4-dimensional Euclidean space in terms of the invariants and covariants on curvature.

Free Research Field

微分幾何学