Locally homogeneous Kaehler manifolds and Transformation groups

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03284
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Josai University
• Principal Investigator: Kamishima Yoshinobu 城西大学, 理学部, 特任教授 (10125304)
• Co-Investigator(Kenkyū-buntansha): 長谷川 敬三 新潟大学, 人文社会科学系, フェロー (00208480)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 幾何構造 / 群の対称性 / 非球形多様体 / 幾何的剛性 / 可微分剛性 / Infra-可解タワー / リー群と等質空間 / 等長群

Outline of Final Research Achievements

The following were the subjects of my research. I.Structure of Isometry groups with radical,and aspherical Riemannian manifolds with large symmetry. Classification of infra-solv tower of fiber bundles.
(Isometry groups with radical,and aspherical Riemannian manifolds with large symmetry.II.Isometric classification of compact locally homogeneous aspherical
Kaehler, Sasaki manifolds. We proved every compact aspherical Riemannian manifold admits a canonical series of orbibundle structures with infra-solv fibers which is called an infra-solv tower. Its length and the geometry of its base measure the degree of continuous symmetry of an aspherical Riemannian manifold. We show that the manifold has large local symmetry if it admits a tower of orbibundle fibrations with locally homogeneous fibers infra-solv tower whose base is a locally homogeneous space. We constructed examples of aspherical manifolds with large local symmetry, which do not support any locally homogeneous Riemannian metrics.

Free Research Field

幾何学とトポロジー

Academic Significance and Societal Importance of the Research Achievements

研究成果の社会への発信は東京という地理的条件もあり大学をあげて努めた.具体的には城西大学紀尾井町キャンパスにおいてコロキュウムを開催,また坂戸(埼玉)キャンパスではオープンユニバーシティでわかりやすく研究成果の一端を社会に還元している.一方で海外には研究集会(サマースクール（Hamburg)を含む）に赴き長期のスパンでの講義・連続講演を提供することで,社会における基盤としての数学の重要性を世界に伝えている.この分野における学術的な意義として,様々な分野への結果に対する,理論的担保と永久の信頼性を与える数学的基盤の構築を行った.