Topology of conformally flat Lorentz manifold and various geometric structures

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540087
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Josai University (2014); Tokyo Metropolitan University (2012-2013)
• Principal Investigator: KAMISHIMA Yoshinobu 城西大学, 理学部, 教授 (10125304)
• Co-Investigator(Kenkyū-buntansha): 長谷川 敬三 新潟大学, 人文社会・教育科学系, 教授 (00208480) ; 相馬 輝彦 首都大学東京, 理工学研究科, 教授 (50154688)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 擬リーマン幾何学 / ローレンツ幾何学 / 微分トポロジー / Developing / Holonomy / Uniformization / 平坦

Outline of Final Research Achievements

When the curvature form for the Cartan connection of the parabolic geometry vanishes, a geomeric manifold has a flat structure. The Riemannian flat manifolds have been studied for a long time. In this note we study conformal Lorentz structure as a pseudo-Riemannian structure. If the Weyl conformal curvature tensor vanishes for a Lorentz manifold, it is called a conformally flat Lorentz manifold. We observed that its geometry and topology. We have shown the following results which never show up in Riemannian geometry. (1) If M is a compact complete Lorentzian similarity manifold, then M is a Lorentzian flat space form.

Free Research Field

幾何学とトポロジー