pursuing global essences of manifolds with geometric structures via special surgeries

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05236
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Hokkaido University
• Principal Investigator: Adachi Jiro 北海道大学, 理学研究院, 研究院研究員 (20374184)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 接分布構造 / Goursat構造 / トーラス手術 / ホモトピー原理

Outline of Final Research Achievements

A geometric structure called a tangent distribution on a manifold is sometimes locally equivalent to the canonical structure on the jet space, that is classically used to deal with differential equations. However, considering it globally, its existence is closely related to the topology of the manifold. In addition, even structures of the same type on a single manifold might be rigid or flexible. In this research project, I developed the torus surgery for manifolds with the Goursat structures and investigated the flexibility of the structures by the method. Furthermore, for some other tangent distributions, I studied the necessary and sufficient condition for a manifold to admit them and the classification of them.

Free Research Field

微分トポロジー

Academic Significance and Societal Importance of the Research Achievements

多様体上の接分布構造とは，多様体と呼ばれる図形の各点に接空間の部分空間を対応させる幾何構造です．その中でも接触構造と呼ばれるものは，古典力学や光学にも関連し古くから研究されてきました．それのみならず，現代の物理学とも密接に関連しています．本研究の成果は接触構造に類似またはその一般化にあたる構造に対し，接触構造の研究と類似の手法の適用の可能性を示すものです．また逆に別の視点からの研究を接触構造の研究にフィードバックする可能性も示します．この研究成果が新たな研究分野の存在を他の研究者に知らしめた一面もあります．