1999 Fiscal Year Final Research Report Summary
Research for manifolds with conformal structure
Project/Area Number 
09440044

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  FUKUOKA UNIVERSITY 
Principal Investigator 
SUYAMA Yoshihiko Fukuoka Univ., Fac. Sci., Prof. > 福岡大学, 理学部, 教授 (70028223)

CoInvestigator(Kenkyūbuntansha) 
KUROSE Takashi Fukuoka Univ., Fac. Sci., Assoc. Prof., 理学部, 助教授 (30215107)
AKUTAGAWA Kazuo Fukuoka Univ., Fac. Sci., Assoc. Prof., 理学部, 助教授 (80192920)
SHIOHAMA Katsuhiro Fukuoka Univ., Fac. Sci. & Engin., Prof., 理工学部, 教授 (20016059)
INOGUCHI Junichi Fukuoka Univ., Fac. Sci., Assist. Prof., 理学部, 助手 (40309886)
YAMADA Kataro Fukuoka Univ., Fac. Sci., Assoc. Prof., 理学部, 助教授 (10221657)

Project Period (FY) 
1997 – 1999

Keywords  conformal structure / conformally flat hypersurface / conformally flat manifold / statistical manfold / conformalprojective transformation / constant mean curvature surface / harmonic map / surface with harmonic inverse mean curvature 
Research Abstract 
1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere. 2. Conformalprojective transformations of statistical manifolds. In this study, we obtained the following result : A conformalprojective transformation of a statistical manifold leaves all umbilical points and the skewsymmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the co
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nformalprojective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformalprojective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold. 3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a noncomplete Riemannian space with nonpositive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3dimensional hyperbolic space form. 4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of Hsurfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gaugetheoretic equation for Hsurfaces with a certain condition of reduction. Less

Research Products
(12 results)