2001 Fiscal Year Final Research Report Summary
On the Weyl conformal invariance on manifolds with various geometric structures and its vanishing of the invariant
| Project/Area Number |
12640082
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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 | Department of Mathematics, Tokyo Metropolitan University |
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Principal Investigator |
KAMISHIMA Yoshinobu Departement of Mathematics, Tokyo Metropolitan Universisty,Professor of Mathematics, 理学(系)研究科(研究院), 教授 (10125304)
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| Co-Investigator(Kenkyū-buntansha) |
GUEST Martin Tokyo Metropolitan Universisty, Professor, 理学(系)研究科(研究院), 教授 (10295470)
OKA Mutsuo Tokyo Metropolitan Universisty, Professor, 理学(系)研究科(研究院), 教授 (40011697)
OHNITA Yoshihiro Tokyo Metropolitan Universisty, Professor, 理学(系)研究科(研究院), 教授 (90183764)
IMAI Jun Tokyo Metropolitan Universisty, Associate Professor, 理学(系)研究科(研究院), 助教授 (70221132)
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| Project Period (FY) |
2000 – 2001
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| Keywords | QCC-structure / Uniformization / Holonomy group / Developing map / Spherical CR Geometry / C-R curvature tenso / Parabolic geometry / Quaternionic hyperbolic geometry |
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| Research Abstract |
1. Bochner curvature flat locally conformal Kahler manifolds: The analogue of Weyl curvature tensor on Kahler manifolds is called the Bochner curvature tensor. As it is a local invariant, the definition makes sense on locally conformal Kahler manifolds (l.c. K. manifolds). We have classified compact Bochner flat Kahler manifolds several years ago. In this continuation, we classified more generally noncompact Bochner flat l.c. K. manifolds.
2. Symmetry and Global rigidity: When there exists a closed noncompact geometric flow, called Lee-Cauchy-Riemann transformations on a compact l.c. K.manifold M, we have shown a rigidity that M will be isometric to the Hopf manifold of standard type.
3. Quaternionic Carnot-Caratheodory structure: We introduced a quaternionic C-C structure on (4n+3)-manifold M and constructed a curvature tensor T which is conformal invariant w. r. t. that geometric structure. If the curvature T vanishes, then we proved that M is locally modelled on the spherical pseudo-quaternionic geometry (Aut_<QC>(S^<4n+3>), S^<4n+3>). In particular, we have established the parabolic geometry on the boundary of the compactification of noncompact semisimple symmetric space of rank 1
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