Canonical forms in geometry and its applications

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05128
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto Institute of Technology
• Principal Investigator: Ikawa Osamu 京都工芸繊維大学, 基盤科学系, 教授 (60249745)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 対称空間 / 超極作用 / Hermann作用 / 対称三対

Outline of Final Research Achievements

・For a given compact connected Lie group and an involution on it, we can define a hyperpolar action, which is called a σ-action. We studied the orbit space and the properties of the action using a symmetric triad. The result is a natural extension of maximal torus theory.
・There exists a one to one correspondence between the set of compact symmetric triads and that of pseudo-Riemannian symmetric pairs, which is a generalization of Cartan’ duality which state a one to one correspondence between the local isomorphic classes of Riemannian symmetric spaces of compact type and the isometric classes of Riemannian symmetric spaces of noncompact type. Thus we call it the generalized duality. We found the applications of generalized duality to Wirtinger inequality and backward Wirtinger inequality.

Free Research Field

数物系科学　数学　幾何学

Academic Significance and Societal Importance of the Research Achievements

超極作用，特にHermann作用，は重要な研究対象であるが，これまで詳しく調べられていたのはコンパクト対称空間へのイソトロピー群の作用であった．コンパクト対称空間へのイソトロピー群の作用の拡張であるHermann作用について詳しく調べることは，今後の幾何学における標準形理論の発展の基礎になる．