Differential geometry on conformal structures and projective structures

• Project/Area Number: 20540084
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Kumamoto University
• Principal Investigator: KOBAYASHI Osamu 熊本大学, 大学院・自然科学研究科, 教授 (10153595)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 微分幾何 / 山辺不変量 / スカラー曲率 / ゲージ理論 / 正則ホモトピー / アファイン接続 / リッチ曲率 / 共形構造 / 射影構造 / Schwarz微分 / 単射性 / 共形微分幾何

Research Abstract

On a curve in the sphere a projective structure is induced from conformal structure of the ambient sphere. It has been shown that if the projective developing map of the curve is injective the curve has no self-intersection. I tried to find other spaces than the sphere on which the injectivity theorem holds. Among compact rank one symmetric spaces only the sphere has this injectivity property, which is the main result of this research project. This theorem with other observations suggests that among all compact Riemannian manifolds only the sphere may have this injectivity property. This will be a future problem.

Research Products

• [Presentation] スカラー曲率に関する未解決と思われる問題について (2009)
  • Place of Presentation: 東北大学
  • Year and Date: 2009-11-09
• [Remarks] 大学院学生セミナー用資料の形にまとめたテキストの草稿を公開している.将来何らかの形で出版される事を見込んでいる.正式なものとは見なされないが,研究活動の証しとして,ここ半年にアップロードしたものを記す.
• [Remarks]
  • URL: http://www.sci.kumamoto-u.ac.jp/~ok/
• [Remarks]
  • URL: http://www.sci.kumamoto-u.ac.jp/~ok/
• [Remarks]
  • URL: http://www.sci.kumamoto-u.ac.jp/~ok/
• [Remarks]
  • URL: http://www.sci.kumamoto-u.ac.jp/~ok/
• [Remarks]
  • URL: http://www.sci.kumamoto-u.ac.jp/~ok/