2006 Fiscal Year Final Research Report Summary
Geometry of almost Hermitian manifolds
| Project/Area Number |
17540068
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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 | Kanazawa University |
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Principal Investigator |
SATO Takuji Graduate School of Natural Science and Technology, Professor, 自然科学研究科, 教授 (30019781)
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| Project Period (FY) |
2005 – 2006
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| Keywords | almost Hermitian manifold / almost Kahler manifold / Kahler structure / holomorphic sectional curvature / tangent bundle |
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| Research Abstract |
By using a 1-parameter family of symmetric affine connections so called α- connections in a statistical model in information geometry, we can introduce a 1-parameter family of almost Kahler structures on the tangent bundle over its statistical model.
Especially, for the case of normal model and discrete distributions model, we studied the almost Kahler structures on then tangent bundles. Our main results are as follows:
1. Almost Kahler structure on the tangent bundle is Kahler iff α = ± 1 (in this case, α-connection is flat.)
2. For the normal model, when α =-1, the Kahler structures on the tangent bundle has constant holomorphic sectional curvature-2, so it is Einstein. On one hand, it is not Einstein when α =1.
3. For the 2 dimensional discrete model case, when α =1, the Kahler structures on the tangent bundle has constant holomorphic sectional curvature 1, so it is Einstein. On one hand, it is not Einstein when α = -1. Further we obtain the followings as a generalization of these results:
4. The almost Kahler structures on the tangent bundle defined by a-connections in the n-dimensional half space with Poincare metric become Kahler iff α=±1. When α =-1, the Kahler structure has constant holomorphic sectional curvature.
5. The almost Kahler structures on the tangent bundle defined by α-connections in the positive part of n-dimensional sphere of radius c become Kahler iff α=±c^2. When α =c^2, the Kahler structure has constant holomorphic sectional curvature.
6. The above result is still hold for the n-dimensional hyperbolic space.
We hope that these results give new point of view in the relation of almost Hermitian geometry and information geometry.
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Research Products
(2 results)
