Geometric analysis on asymptotically symmetric spaces and parabolic geometries

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K14189
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: Matsumoto Yoshihiko 大阪大学, 理学研究科, 助教 (00710625)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 微分幾何学 / リーマン幾何学 / アインシュタイン方程式 / 漸近的対称空間

Outline of Final Research Achievements

The hyperbolic space, a mathematical concept found in the nineteenth century that opened the door toward modern study of geometry, has been generalized in many ways, and there is a relatively recent generalization called asymptotically symmetric spaces. Of central interest are those spaces satisfying the Einstein equation.
Our aim in this research project was to deliver substantial progress in this field. We proposed that asymptotically symmetric spaces of a certain type be furnished with an additional geometric structure (an almost complex structure satisfying some partial differential equation), and proved a theorem that systematically provides examples of such a furnishment. Moreover, we obtained a new construction of a family of asymptotically symmetric spaces of another type, and studied the interesting limiting behavior of the family.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

本課題で実施した研究は純粋数学に属するもので、近い未来に実用的な意味で社会に役立つことはおそらくない。しかし、人類の共有する知的地平を広げるという点において意義がある。ひいては、国際社会において日本が文化的に敬意をもたれることにも、多少の貢献があるかもしれない。
学術的には、世界的にみて類例のない研究の方向性を示したものと信ずる。この方向性を発展させることができるか、引き続き取り組みを進めていきたい。また、物理学には何らかの形で影響をもたらす可能性があると期待される。