Project/Area Number  10640088 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  Saga University 
Principal Investigator 
CHENG Qingming Saga University, Professor, 理工学部, 教授 (50274577)

CoInvestigator(Kenkyūbuntansha) 
YAMASAKI Masayuki Josai University, Professor, 理学部, 教授 (70174646)
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)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 2001 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 2000 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1999 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  Scalar curvature / Enclidean space / Complete submanifold / Sphere / Mangoldt Surface / radial curvature / mean curvature vector / AlexandrovTopologov Theorem / scalar curvature / Euclidean space / complete submanifold / sphere / Mangoldt surface / AlexandrovToponogov theorem / Ricci curvature / principal curvature / hypersurface / Riemannian manifold / Conformally flat / Riemannian product / Submanifolds / local conformallyflat / minimal hypersurfaces / submanifolds / Clifford torus / constant scalar curvature / second fundamental form / conformally flat 
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 3dimensional complete locally conformally flat Riemannian manifolds with nonnegative 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 codimension. 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 AlexandrovTopologov theorem.
