2004 Fiscal Year Final Research Report Summary
Research on curvature structures and topological structures of submanifolds in Riemannian manifolds
| Project/Area Number |
14540085
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Saga University |
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Principal Investigator |
CHENG Qing-ming Saga Univ., Fac.of Sci.and Eng., Prof, 理工学部, 教授 (50274577)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOHAMA Katsuhiro Saga Univ., Fac.of Sci.and Eng., Prof, 理工学部, 教授 (20016059)
ISHIKAWA Susumu Saga Univ., Fac.of Sci.and Eng., Prof, 理工学部, 教授 (10039258)
KAWAI Shigeo Saga Univ., Fac.of Sci.and Eng., Prof, 文化教育学部, 教授 (30186043)
MATSUZOE Hiroshi Saga Univ., Fac.of Sci.and Eng., Ass.Prof, 理工学部, 助教授 (90315177)
MASHIKO Yukihiro Saga Univ., Fac.of Sci.and Eng., Lecture, 理工学部, 講師 (00315178)
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| Project Period (FY) |
2002 – 2004
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| Keywords | complete submanifod / mean curvature / scalar curvature / eigen values of Laplacian / Ricci curvature / sectional curvature / Euclidean space and sphere / homogeneous manifold |
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| Research Abstract |
In this project, we mainly investigated the curvature structures and topological structures of submanifolds in Riemannian manifolds and the geometry of eigenvalues and eigenfunctions of Laplacian on Riemannian manifolds. It is our purpose to research geometric problems on properties oftopology and curvatures of many kinds of manifolds by means of many different methods. We studied (1)the geometry of topological structures and curvature structures of submanifolds in Euclidean spaces and spheres. (2)the geometry of compact hypersurfaces with infinite fundamental group in spheres. (3)the geometry of curvature structures of complete hypersurfaces in 4-dimensional space forms. (4)the geometry of curvature structures of complete space-like hypersurfaces in Lorentz spaces. (5)the geometry ofeigenvalues and eigenfunctions of Laplacian on Riemannian manifolds. (6)the geometry of sphere theorems. (7)the geometry of conformally projective structures of satistical manifolds. (8)the geometry on radial curvature and topology of manifolds and obtained many important results.
It is characterizations of our project that important contributions on the research of topological structures and curvature structures of complete submanifolds in Riemannian manifolds are obtained. Furthermore, Cheng and Yang obtained important results for the investigation on eigenvalues of Laplacian on Riemannian manifolds.
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