Metric invariants and space structures
| Project/Area Number |
17540079
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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 | Okayama University of Science |
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Principal Investigator |
SAKAI Takashi Okayama University of Science, FACULTY OF SCIENCE, PROFESSOR (70005809)
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| Co-Investigator(Kenkyū-buntansha) |
KIYOHARA Kazuyoshi OKAYAMA UNIVERSITY, FACULTY OF SCIENCE, PROFESSOR (80153245)
ITOH Jin-ichi KUMAMOTO UNIVERSITY, FACULTY OF EDUCATION, PROFESSOR (20193493)
KATSUDA Atsushi OKAYAMA UNIVERSITY, FACULTY OF SCIENCE, ASSOCIATE PROFESSOR (60183779)
MORI Yoshiyuki OKAYAMA UNIVERSITY OF SCIENCE, FACULTY OF SCIENCE, LECTURER (00388919)
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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 |
¥3,700,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Riemannian manifolds / Distance functions / Morse theory / Geodesics / Cut loci |
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| Research Abstract |
Distance function d_p to a point p of a compact Riemannian manifold is an important geometric function that is also related to the manifold structure. However, distance function d_p admits a point where d_p is not differentiable, and such a point is contained in the cut locus C_p of p. It was known that the notion of critical points may be introduced as in usual Morse theory and d_p is not differentiable at critical points. In the case where there are no critical points, Morse theory (the isotopy lemma) has been developed as in smooth case that played an important role in applications.
Now to develop Morse theory for distance functions including critical points, one should first define the notion of the index at a critical point. Under the support of the present Grant-in-Aid, T. Sakai, head investigator of this research program, carried out this with J. Itoh when Riemannian metric satisfies some natural non-degeneracy conditions, and developed Morse theory for distance functions from a
… More
direct geometric view point (has been published in a Journal). There it is important to show that the cut locus Cp carries a nice structure (Whitney stratification), and the index of a critical point q is defined in terms of the number of minimal geodesics joining p, q and the usual index at q of the restriction of 4 to the stratum containing q that is a smooth function with a critical point q. I hope to continue to study how generic is the conditions among all Riemannian metrics on a given manifold.
With respect to distance functions, K. Kiyohara determined the explicit structure of the cut locus and the conjugate locus of any point in ellipsoids (and some Liouville manifolds) from a view point of integrable geodesic flow with J. Itoh, and A. Katsuda studied the inverse problem of the Neumann boundary value problem, namely how to reconstruct the inner Riemannian metric from the distance function to the boundary of a Riemannian manifold with boundary.
As for geometric inequalities (isosystolic inequality, isodiametric inequality) and the first eigenvalue estimate of the Laplacian of a compact Riemannian manifolds with non-negative Ricci curvature and its perturbation, we could not make substantial progress in this period, but Kiyohara obtained a result concerning an inequality of geometric invariants of Alexandrov spaces. I hope to continue to carry the program. Y. Mori investigated algorithm of quantum computer and supported Sakai in computer aid. Less
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Report
(4 results)
Research Products
(153 results)
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[Presentation] Every point is critical2006
Author(s)
J. Itoh, I. Barany, C. Vilcu, T. Zamfirescu
Organizer
Autumn meeting of Math. Soc. Japan
Place of Presentation
Osaka City Univ
Year and Date
2006-09-20
Description
「研究成果報告書概要(欧文)」より
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[Presentation] 閉じていない空間曲線の絶対全曲率2006
Author(s)
榎本 一之, 伊藤 仁一, R.Sinclair
Organizer
日本数学会年会幾何学分科会
Place of Presentation
中央大学
Year and Date
2006-03-26
Description
「研究成果報告書概要(和文)」より
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[Presentation] リウヴィル多様体のカットローカス2006
Author(s)
清原 一吉
Organizer
「測地線および関連する諸問題」研究集会
Place of Presentation
熊本大学教育学部
Year and Date
2006-01-08
Description
「研究成果報告書概要(和文)」より
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[Presentation] 距離関数のモース理論2005
Author(s)
酒井 隆
Organizer
第3回岡山幾何学談話会
Place of Presentation
岡山理科大学
Year and Date
2005-06-24
Description
「研究成果報告書概要(和文)」より
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[Book] 解決!ポアンカレ予想(数学セミナー増刊)2007
Author(s)
小島定吉, 糸川銚, 酒井隆, 戸田正人, 小林治, 二木昭人, 浦川肇, 十文字正樹, 山口孝男, 塩谷隆, 小林亮一
Total Pages
94
Publisher
日本評論社
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