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2001 Fiscal Year Final Research Report Summary

The Study of geometry of localry conformally flat Riemannian manifolds

Research Project

Project/Area Number 10640088
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionSaga University (2001)
Josai University (1998-2000)

Principal Investigator

CHENG Qing-ming  Saga University, Professor, 理工学部, 教授 (50274577)

Co-Investigator(Kenkyū-buntansha) YAMAGUCHI Hiroshi  Josai University, Professor, 理学部, 教授 (20137798)
ISHIKAWA Susumu  Saga University, Professor, 理工学部, 教授 (10039258)
SHIOHAMA Katsuhiro  Saga University, Professor, 理工学部, 教授 (20016059)
YAMASAKI Masayuki  Josai University, Professor, 理学部, 教授 (70174646)
Project Period (FY) 1998 – 2001
KeywordsScalar curvature / Enclidean space / Complete submanifold / Sphere / Mangoldt Surface / radial curvature / mean curvature vector / Alexandrov-Topologov Theorem
Research Abstract

In this project, we mainly investigated the geometry of complete locally conformally flat Riemannian manifolds and the geometry of submanifolds. It is our purpose to research geometric problems on topoiogical structures and curvature structures of many kinds of manifolds by many different methods. We studied (1) the geometry of conformally flat manifolds, (2) the geometry of hypersurfaces in space forms, (3) the geometry of submanifolds, (4) the geometry of sphere theorems and (5) the geometry of Alexandrov spaces and obtained many new results.
On the research of (1), we classified 3-dimensional complete locally conformally flat Riemannian manifolds with non-negative constant scalar curvature and constant norm of the Ricci tensor. We also gave certain characterizations of such manifolds with negative constant scalar curvature.
On the research of (2), we proved that complete hypersurfaces in a Euclidean space with constant scalar curvature and two distinct principal curvatures are isometric to complete and noncompact hypersurfaces of revolution. From this result, we obtained a classification of complete locally conformally fiat hypersurfaces in a Euclidean space with constant scalar curvature and gave a partial answer of Yau's conjecture.
On the research of (3), we extended the result due to Klotz and Osserman on complete surfaces in the 3dimensional Euclidean space with constant mean Curvature to any higher dimension and any higher co-dimension. We also obtained important result on complete submanifolds in the Euclidean space with constant scalar curvature.
On the research of (4), we proved that under condition of radial curvature, maximal diameter theorem holds.
On the research of (5), we obtained a new version of Alexandrov-Topologov theorem.

  • Research Products

    (39 results)

All Other

All Publications (39 results)

  • [Publications] Qing-Ming Cheng: "Chern's Conjecture on minimal hypersurfaces"Math. Zeit.. 227. 377-390 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "The relation between degree and dilation of continuous"Japanese Math. Jour.. 24. 183-190 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Space like hypersurfaces in de Sitter spaces with constant scalar curvature"Manuscripta Math.. 95. 499-505 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "A Characterization of the Clifford torus"Proc. Amer. Math. Soc.. 127. 819-828 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Complete maximal spacelike surfaces in anti-de Sitter space H^4_2(c)"Glasgow Math. J.. 42. 139-156 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Hypersurfaces in Euclidean spaces"Proce. Fifth Pacific Rim Geom. Conference Tohoku Univ. Sendai, Japan. Tohoku Math. Publ.. 20. 33-42 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Complete submanifolds in Euclidean spaces with Parallel mean curvature vector"Manuscripta Math.. 105. 353-366 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Compact locally conformally flat Riemannian manifolds"Bull. London Math. Soc.. 33. 459-465 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Conformally flat 3-manifolds with constant scalar curvature II"J. Japanese Math.. 27. 387-404 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Hypersurfaces in a unit sphere S^<n+1>(1) with constant scalar curvature"J. London Math. Soc.. 64. 755-768 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Submanifolds with constant scalar curvature"Proc. Royal Society Edinbergh. 131A. (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Complete hypersurfaces in a Euclidean space R^<n+1> with constant scalar curvature"Indiana Univ. Math. J.. (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhiro Shiohama: "The unboundedness of certain minimal sub-manifolods of positively curved Riemannian spaces"Diffe. Geom. and Its Applications. 11. 97-104 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhiro Shiohama: "Rigidity and sphere theorems for submanifolds II"Kyushu J. Math.. 54. 103-109 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhiro Shiohama: "Comparison theorems for manifolds with radial curvature bounded below"Josai Math. Monograghs. 3. 81-91 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhiro Shiohama: "Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below"Math. Z.. (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhiro Shiohama: "Generalized Toponogov's theorem for manifolds with radial curvature bounded below"Nagoya Math. J.. (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hiroshi Yamaguchi: "On the product of Riesz sets in dual Objects of compact groups"Transa. of a Japanese-German Symposium. 360-372 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Masayuki Yamasaki: "Controlled Surgery Theory"SUGAKU EXPOSITIONS. 13. 113-124 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Katsuhiro Shiohama: "Handbook of Differential Geometry"Elsevier. 1054 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng: "Differential Geometry"Josai University. 154 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Qing-Ming Cheng and Hongchang Yang: "Chern's Conjecture on minimal hypersurfaces"Math. Zeit.. 227. 377-390 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-MIng Cheng, Yoshiro Machigashira and Yoshihiro Sugioka: "The relation between degree and dilation of continuous maps"Japanese Math. Jour.. 24. 183-190 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng and Susumu Ishikawa: "Space like hypersurfaces in de Sitter spaces with constant scalar curvature"Manuscripta Math.. 95. 499-505 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng and Susumu Ishikawa: "A Characterization of the Clifford torus"Proc. Amer. Math. Soc.. 127. 819-828 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng and Susumu Ishikawa and Katsuhiro Shiohama: "Conformally flat 3-manifolds with constant scalar curvature"J. Math. Soc. Japan. 51. 209-226 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng: "Complete maximal spacelike surfaces in anti-de Sitter space H^∧4_2(c)"Glasgow Math.l J.. 42. 139-156 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng: "Hypersurfaces in Euclidean spaces, Proc. Fifth Pacific Rim Geom. Conference"Tohoku Univ., Sendai, Japan TohokuMath. Publ.. 20. 33-42 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng and Kazuhiro Nonaka: "Complete submanifolds in Enclidean spaces with parallel mean curvature vector"Manuscripta Math.. 105. 353-366 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng: "Compact locally conformally flat Rikemannian manifolds"Bull. London Math. Soc.. 33. 459-465 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng, Susumu Ishikawa and Katsuhiro Shiohama: "Conformally flat 3-manifolds with constant scalar curvature II"J. Japanese Math.. 27. 387-404 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng: "Hypersurfaces in a unit sphere S^∧{n+1})(1) with constant scalar curvature"J. London Math. Soc.. 64. 755-768 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng: "Submanifolds with constant scalar curvature"Proc. Royal Soc. Edinbergh 31A. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Qing-Ming Cheng: "Complete hypersurfaces in a Euclidean space R^∧{n+1} with constant scalar curvature"the Indiana University Mathematics Journal. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Itokawa, K. Shiohama: "The unboundedness of certain minimal submanifolds of postively curved Riemannian spaces"Diff. Geom.its Appl.. 11. 97-104 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Shiohama and H. Xu: "Rigidity and sphere theorems for submaanifolds II"Kyushu J. Math. 54. 103-109 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. Tanaka and K. Shiohama: "Compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below"Math. Z.. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Itokawa, Y. Machigashira and K. Shiohama: "Generalized Toponogov's theorem for manifolds with radial curvature bounded below"Nagoya Mathematical Journal. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masayuki Yamasaki: "Controlled Surgery Theory"SUGAKU EXPOSITIONS. 13. 113-124 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2003-09-17  

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