Dynkin indices and totally geodesic submanifolds in Riemannian symmetric spaces

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14310
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Hiroshima University
• Principal Investigator: Okuda Takayuki 広島大学, 先進理工系科学研究科(理), 准教授 (40725131)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 等質空間 / 対称空間 / 部分多様体 / 粗幾何学

Outline of Final Research Achievements

Through this research, we successfully defined a natural number called the split Dynkin index for all totally geodesic embeddings between Riemannian symmetric spaces using the theory of Lie algebras. This correspondence generalizes the concept known as the Dynkin index for Lie algebra homomorphisms between complex simple Lie algebras and is considered an important concept for understanding the relationships between Riemannian symmetric spaces. We also developed a method to calculate the corresponding split Dynkin index for each embedding using the concept of complexification. Additionally, during the course of this research, the relationship between discontinuous groups on homogeneous spaces and coarse geometry has been elucidated.
These results are currently being prepared for submission as a journal paper.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

リーマン対称空間と呼ばれる性質を持つ空間は幾何学において基本的かつ重要な研究対象である. 本研究はそれらの間の関係性を理解するためのものである. 本研究においてはリーマン対称空間の間に全測地的はめ込みという関係性が与えられたとき, それをある尺度において自然数で評価する手法を開発できた. この手法を用いて今後リーマン対称空間の関係性をより深く理解することが可能となった.