2007 Fiscal Year Final Research Report Summary
Study of conformal differential geometry
| Project/Area Number |
16540075
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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 | Kumamoto University |
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Principal Investigator |
KOBAYASHI Osamu Kumamoto University, Department of mathematics, professor (10153595)
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| Project Period (FY) |
2004 – 2007
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| Keywords | conformal strucuture / projective structure / affine connection / scalar curvature / Ricci curvature |
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| Research Abstract |
Among many geometric structures of a manifold we are mainly interested in those structures which are closely related to the conformal geometry. Here are some of main results of this research project :
1. For a regular curve x : I>M in a conformal manifold M, we can define a projective structure on the interval. Moreover if M is the standard sphere and if the projective developing map of I to the real projective line is injective, then the curve x is injective
2. Suppose that we are given a complex number a which is not real and that f is a continuously differentiable complex function defined on a domain in the complex plain. If for four points which have the anharmonic ratio a, the images of the four points also have the same anharmonic ratio, then f is a Moebius transformation. This result is an extension of a theorem by Haruki and Rassias in 1996.
3. An analogy of the Yamabe problem in conformal differential geometry is formulated in projective differential geometry. We have given a characterization of the Riemannian connection of an Einstein metric of negative scalar curvature only in terms of affine differential geometry using, the variational method.
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[Book] 21世紀の数学2004
Author(s)
小林 治(宮岡 礼子(編)
Total Pages
392-393
Publisher
日本評論社
Description
「研究成果報告書概要(和文)」より
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