2007 Fiscal Year Final Research Report Summary
Integral geometry in homogeneous spaces and its applications
| Project/Area Number |
18540065
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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 | University of Tsukuba |
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Principal Investigator |
TASAKI Hiroyuki University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor (30179684)
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| Co-Investigator(Kenkyū-buntansha) |
ITOH Mitsuhiro University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor (40015912)
MASHIMO Katsuya Tokyo University of Agriculture and Techinology, Department of Mathematics, Professor (50157187)
IKAWA Osamu Fukushima Natioanl Colledge of Technology, Department of General Education, Associate Professor (60249745)
KOKUBU Masatoshi Tokyo Denki University, Department of Mathematical Sciences, Associate Professor (50287439)
SAKAI Takashi Osaka City University, Graduate School of Sciences, Assistant (30381445)
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| Project Period (FY) |
2006 – 2007
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| Keywords | weakly reflective orbit / austere orbit / Riemannian symmetric pair / kinematic formula / multiple Kaehler angle / integral geometry |
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| Research Abstract |
In the research of integral geometry in the homogeneous space, integration on the orbit of a linear isotoropy action performs the key role. Many of important examples are austere submanifolds. The representative, Ikawa and Sakai classified the austere orbits of linear isotoropy actions of the Riemann symmetry pair in the sphere. When those austere orbits were examined in detail, a lot of austere orbits have not only the symmetry of the second fundamental form, but a certain kind of global symmetry. Because this global symmetry was a character to weaken the definition of reflective submanifold, it was named weakly reflective submanifold and examined its basic property. Weakly reflective submanifolds are austere submanifolds, and austere submanifolds are minimal submanifolds. The weakly reflective orbits was able to be classified in addition to the classification of the above-mentioned austere orbits. It is shown that the orbit with degenerate Gauss map becomes a weakly reflective subman
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ifold, and has generalized this though there was a result that the orbit of cohomogeneity 1 with degenerate Gauss map becomes austere before. That is, orbits with degenerate Gauss map are weakly reflective, and weakly reflective orbits are austere.
The research of the orbits of linear isotoropy action of Riemann symmetry pairs is important to lead various relations concerning kinematic formula and the quermassintegrals in real space forms and complex space forms. Actually, the concept of the multiple Kaehler angle that the representative introduced was able to be led from the viewpoint of geometry of the orbit naturally, and when kinematic formula in complex space forms was formulated, the character to have obtained from geometrical consideration in the orbit played a basic role. In addition, a multiple Kaehler angle and its basic properties are important bases in the re-construction of integral geometry that makes the concept of valuation that has progressed in the past, several years a base. The viewpoint of geometry of the orbit is indispensable to research integral geometry in this direction of the future. Less
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Research Products
(8 results)
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[Presentation] 積分の近似和の収束の速さ2008
Author(s)
田崎博之
Organizer
研究集会「等質空間の不変構造と部分多様体」
Place of Presentation
千葉工業大学(津田沼)
Year and Date
20080313-14
Description
「研究成果報告書概要(和文)」より
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