Study on the finiteness of the fundamental groups of Lorentzian manifolds

• Project/Area Number: 15K17537
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Toyota Technological Institute (2017); Nagoya University (2015-2016)
• Principal Investigator: MUKUNO Junichi 豊田工業大学, 工学部, ポストドクトラル研究員 (50737301)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: マイヤースの定理 / 特異点定理 / 不定値計量 / 基本群 / 測地的完備性 / 曲率テンソル / 大域的双曲型ローレンツ多様体 / 擬リーマン幾何 / ローレンツ幾何 / 曲率 / 等質空間 / 断面曲率

Outline of Final Research Achievements

I have studied whether or not for Lorentzian geometry, and in general pseudo-Riemannian geometry, the positivity of curvature tensor and the geodesic completeness lead to the finiteness of the fundamental group. This is to research an analogy of the Myers theorem in pseudo-Riemannian geometry. I have found that if classes of geodesically complete Lorentzian manifolds and pseudo-Riemannian manifolds have the positivity of curvature tensor, then the fundamental group of some Riemannian submanifold of maximal dimension is finite. I have constructed new examples of pseudo-Riemannian homogeneous spaces with the positivity of curvature tensor.

Research Products

• [Journal Article] On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor (2015)
  • Author(s): Jun-ichi Mukuno
  • Journal Title: Differential Geometry and its Applications Volume: 41 Pages: 33-38
  • DOI: 10.1016/j.difgeo.2015.04.002
  • Peer Reviewed
• [Presentation] On the fundamental group of semi-Riemannian manifolds with positive curvature operator (2017)
  • Author(s): 椋野純一
  • Organizer: Rigidity School Nagoya 2017
  • Int'l Joint Research / Invited
• [Presentation] 正の曲率作用素を持つ擬 Riemann 多様体の基本群について (2017)
  • Author(s): 椋野純一
  • Organizer: 日本数学会 2017年度秋季総合分科会
• [Presentation] On the fundamental group of semi-Riemannian manifolds with positive curvature operator (2017)
  • Author(s): 椋野純一
  • Organizer: The third Japanese-Spanish workshop on Differential Geometry (ポスター発表)
  • Int'l Joint Research
• [Presentation] On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor(ポスター発表) (2016)
  • Author(s): 椋野純一
  • Organizer: 8th International Meeting on Lorentzian Geometry
  • Place of Presentation: マラガ大学(スペイン, マラガ)
  • Year and Date: 2016-09-20
  • Int'l Joint Research
• [Presentation] On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor (ポスター発表) (2015)
  • Author(s): 椋野純一
  • Organizer: International conference Relativity and Geometry in memory of Ande Lichnerowicz
  • Place of Presentation: Institut Henri Poincare (Paris, France)
  • Year and Date: 2015-12-14
  • Int'l Joint Research
• [Presentation] On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor (2015)
  • Author(s): 椋野純一
  • Organizer: Geometry and Analysis of Discrete Groups and Hyperbolic Spaces
  • Place of Presentation: 京都大学数理解析研究所 (京都市)
  • Year and Date: 2015-06-25
  • Int'l Joint Research / Invited
• [Remarks] Jun-ichi Mukuno's Home Page
  • URL: https://sites.google.com/site/junichimukuno/home
• [Remarks] Jun-ichi Mukuno's Home page
  • URL: https://sites.google.com/site/junichimukuno/
• [Remarks] Jun-ichi Mukuno’s Home page
  • URL: https://sites.google.com/site/junichimukuno/