2007 Fiscal Year Final Research Report Summary
Comprehensive studies of cut locus
| Project/Area Number |
17540085
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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 |
ITOH Jin-ichi Kumamoto University, Faculty of Education, Professor (20193493)
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| Co-Investigator(Kenkyū-buntansha) |
HIRAMINE Yutaka Kumamoto Univ., Faculty of Education, Professor (30116173)
KUWAE Kazuhito Kumamoto Univ., Faculty of Education, Ass. Professor (80243814)
SAKAI Takashi Okayama Sci Univ, Faculty of Science, Professor (70005809)
KIYOHARA Eazuhiro Okayama Univ., Faculty of Science, Professor (80153245)
TANAKA Mimoru Tokai Univ., Faculty of Science, Professor (10112773)
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| Project Period (FY) |
2005 – 2007
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| Keywords | geodesies / cut locus / 1'st conjugate locus / Liouville manifold / guadric surface / distance function / farthest point / unfolding of polyhedron |
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| Research Abstract |
We studied the cut loci and related several topics, for example, the farthest points or simple closed geodesies on convex surfaces and got many results.
As the problem to determine the cut locus, we proved that some compact Liouville manifolds have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds. We discussed on an extension to general dimension of Jacobi's last theorem(conjugate loci on ellipsoids have exact four cusps), and we got some remarkable progress and are planning to continue the research. Moreover, as the related topics, we got a modern proof of thread constructions of general quadric(hyper)surfaces by using the first integral (jointed work with K. Kiyohara).
As the problem to study structures of cut locus, under some non-degenerating assumption we proved that the cut locus admits a nice stratification, some cone structure locally. Under stronger assumptions we have simpler procedure of Morse theory by using of critical points of distance functions (jointed work with T. Sakai).
We established, for general convex surfaces, inequalities involving the diameter, the area and the length of simple closed quasi-geodesics (jointed work with C. Vilcu).
Using the above simple closed quasi-geodesics we proved that any polyhedra are unfolded to a planar simple polygon by some cutting (jointed work with J. O'Rourke, C. Vilcu).We discussed several other unfolding by using simple clodes quasi geodesics, also
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Research Products
(93 results)
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[Presentation] Every point is critical2007
Author(s)
J., Itoh(j. w., Imre, Barany, Costin, Vilcu, Tudor Zamfirescu)
Organizer
Conf."Region int. geom. structures and submnifolds"
Place of Presentation
Meijo univ.
Year and Date
2007-03-08
Description
「研究成果報告書概要(欧文)」より
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[Presentation] Every point is critical2006
Author(s)
J., Itoh(j. w., I., Barany, C., Vilcu, T., Zamfirescu)
Organizer
Workshop on Geometric and Topological Combinatoric
Place of Presentation
Alcalade Henares, Spain
Year and Date
2006-09-01
Description
「研究成果報告書概要(欧文)」より
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