2003 Fiscal Year Final Research Report Summary
Geometry of Almost Complex Manifolds
| Project/Area Number |
14540070
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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 | NIIGATA UNIVERSITY |
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Principal Investigator |
SEKIGAWA Kouei NIIGATA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (60018661)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUSHITA Yasuo Shiga Prefectural University, Faculty of Technology, Professor, 工学部, 教授 (90144336)
HASEGAWA Keizo NIIGATA UNIVERSITY, Facalty of Education and Human Science, Associate Professor, 教育人間科学部, 助教授 (00208480)
INNNAMI Nobuhiro NIIGATA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (20160145)
HASHIMOTO Hideya Meijo University, School of Science and Technology, Professor, 理工学部, 教授 (60218419)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Almost complex manifold / Integrability / (Almost) Kahler manifold / Einstein metric / Goldberg conjecture / Nearly Kahler 6-dimensional sphere / J-holomorphic curve / CR-submanifold |
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| Research Abstract |
A smooth manifold M admitting (1,1) tensor field J satisfying J^2=-I is called an almost complex manifold. The concept of almost complex manifold is a natural generalization of complex manifold. There are known many examples of almost complex manifolds which are not complex manifolds. It is well-known that 6-dimensional sphere S^6 admits an almost complex structure which is not complex one. Almost complex manifold(M,J) is said to be integrable if M admits a complex structure and the associated almost complex structure coincides with the almost complex structure J.In the research project, we considered mainly the following topics in Almost Complex Geometry :
(1)The Goldberg conjecture
(2)Submanifolds in nearly Kahler 6-dimensional sphere S^6
(3)Related topics to the above (1) and (2)
Concerning (1),we obtained some affirmative answers to the conjecture under some additional curvature conditions in 4-dimensional case. Recently, Y.Matsushita found a counter example to the indefinite-version of the conjecture in dimension 4.Concerning (2),we obtained some topological conditions for a given 4-dimensional submanifold of S^6 to be CR-submanifold and constructed several examples of 4-dimensional CR-submanifolds of S^6. On one hand, H.Hashimoto(and et al.) constructed J-holomorphic flat tori of type(III)in S^6 and classified them from the integral system view point. Concerning (3),we examined local structure of Kahler surfaces with distinct constant Ricci eigenvalues and determind all homogeous Kahler surfaces. We note that the existence of compact KShler surface with distinct constant negative scalar curvatures is deeply concerned with the possibility of existence of counter example of the Goldberg conjecture. Further, K.Hasegawa proved that a 4-dimensional compact solvmanifold admitting Kahler structure is holomorphically equivalent to a torus bundle over a torus with a certain finite abelian structure group.
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