| Project/Area Number |
17340018
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Kyushu University |
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Principal Investigator |
NAGATOMO Yasuyuki Kyushu University, Graduate School of Mathematics, Associate Professor (10266075)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Kotaro KYUSHU UNIVERSITY, Graduate School of Mathematics, Professor (10221657)
ITOH Mitsuhiro University of Tsukuba, Institute of Mathematics, Professor (40015912)
OHNITA Yoshihiro Osaka City University, Department of Mathematics, Professor (90183764)
TASAKI Hiroyuki University of Tsukuba, Institute of Mathematics, Associate Professor (30179684)
TAKAYAMA Shigeharu Tokyo University, Graduate School of Mathematics, Associate Professor (20284333)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥6,490,000 (Direct Cost: ¥5,800,000、Indirect Cost: ¥690,000)
Fiscal Year 2007: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2006: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2005: ¥2,200,000 (Direct Cost: ¥2,200,000)
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| Keywords | ASD connections / vector bundles / moduli spaces / quaternion manifolds / Lie groups / twistor spaces / vanishing theorems / harmonic mans / ASD接続 / コホモロジー / 消滅定理 |
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| Research Abstract |
In 2005, we focus our attention on submanifolds which appear as singular sets of ideal instantons. Those are zero loci of the twistor sections satisfying linear equations which are linearization of the (higher dimensional instanton equations. Moreover, we construct embeddings of the Wolf spares into Grassmannian_, which turn out to be minimal embeddings. We also obtain vanishing theorems for cohomology groups.
In 2006, we succeeded to find some relations between harmonic mappings into Grasmannians and the Yang-Mills connections, which are essential and important steps to our subject. We obtain a condition for a map of a Riemannian manifold into Grassmannian to be a harmonic map. We use this condition to obtain the classification of harmonic maps with constant energy density from holonomy irreducible homogeneous manifold s into Grassmannian manifolds. In addition, we can show that a vector bundles on a real Grassmannian manifold with some topological type admits a unique ASD connection u
… More
p to gauge equivalence.
In 2007, we consider the cases that a harmonic map into Grassmannian is a totally geodesic one. As a result, we obtain the classification of totally geodesic immersions of irreducible type. In this classification, we obtain an integral formula which indicates the dimension of Grassmannian, which is the target space of the mapping. In the case of the complex projective line, we can show that an indecomposable totally geodesic immersion is an totally geodesic immersion of the irreducible type. To obtain the result, we use the above characterization of a harmonic map and construct a variant of the spherical function theory on homogeneous vector bundles. This implies that we can classify all totally geodesic immersions of complex projective line into Grassmannians. We develop an analogue of the "geometry of the twistor sections" on symmetric spaces of compact type. This gives us pairs of totally geodesic submanifolds on almost symmetric spaces of compact type. These pairs are intimately related to vector bundles and sections of them. Indeed, we can construct a function using a section, which is an isoparametric function on every Grassmann manifold. This function gives a family of submanifolds as level sets. We can find one and only minimal submanifold in this family. Less
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