Integral geometry in homogeneous spaces and its applications

均匀空间中的积分几何及其应用

• 批准号: 18540065
• 负责人: TASAKI Hiroyuki
• 金额: $ 2.5万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540065/
• 关键词: weakly reflective orbit; austere orbit; Riemannian symmetric pair; kinematic formula; multiple Kaehler angle; integral geometry; Riemann等質空間; Riemann対称空間; Croftonの公式; 鏡映部分多様体; 弱鏡映部分多様体; austere部分多様体

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 … More 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

在齐次空间积分几何的研究中，线性等规作用量轨道上的积分起着关键的作用。许多重要的例子是严格子流形。代表人物Ikawa和Sakai对球面上黎曼对称对的线性同构作用的严格轨道进行了分类。当我们仔细研究这些严格轨道时，发现许多严格轨道不仅具有第二基本形式的对称性，而且具有某种整体对称性。由于这种整体对称性是弱化反射子流形定义的一个特征，我们称之为弱反射子流形，并研究了它的基本性质。弱反射子流形是严格子流形，严格子流形是极小子流形。弱反射轨道是除了上述的严峻的轨道的分类能够被分类。证明了具有退化高斯映射的轨道成为弱反射子轨道 ...更多信息 ifold的结果，并将这一结论推广到了退化Gauss映射的余齐性1的轨道变得严峻的情况。Riemann对称对的线性同构作用的轨道的研究对于导出真实的空间形式和复空间形式中的运动学公式和准质量积分的各种关系具有重要意义。实际上，该代表所提出的多重Kaehler角的概念可以从轨道几何的观点自然地导出，而在制定复空间形式中的运动学公式时，从轨道的几何考虑所获得的性质起着基础性的作用。此外，多重Kaehler角及其基本性质是重建积分几何的重要基础，这使得过去几年取得进展的赋值概念成为基础。轨道几何的观点是今后研究积分几何这一方向不可缺少的。少

项目成果

積分の近似和の収束の速さ

近似积分和的收敛速度

• 发表时间: 2008
• 作者: Kokubu;Masatoshi; Rossman;Wayne; Umehara;Masaaki; Yamada;Kotaro;Mitsuhiro Itoh;Masatoshi Kokubu;Hiroyuki Tasaki;Katsuhiro Moriya;Katsuya Mashimo;Koji Tojo;田崎博之
• 通讯作者: 田崎博之

On the branching theorem of the pair (F_4, Spin (9) )

关于偶对的分支定理 (F_4, Spin (9) )

• 发表时间: 2006
• 期刊: Tsukuba J. Math. 30
• 作者: U-Hang Ki;Setsuo Nagai;Ryoichi Takagi;Katsuya Mashimo
• 通讯作者: Katsuya Mashimo

Flat fronts in hyperbolic 3-space and their caustics

• DOI: 10.2969/jmsj/1180135510
• 发表时间: 2005-11
• 期刊: Journal of The Mathematical Society of Japan
• 影响因子: 0.7
• 作者: M. Kokubu;W. Rossman;M. Umehara;Kotaro Yamada

A space of minimal tori with one end and cyclic symmetry

具有一端且循环对称的最小环面空间

• 发表时间: 2006
• 期刊: Tsukuba Journal of Mathematics 30・1
• 作者: Kokubu;Masatoshi; Rossman;Wayne; Umehara;Masaaki; Yamada;Kotaro;Mitsuhiro Itoh;Masatoshi Kokubu;Hiroyuki Tasaki;Katsuhiro Moriya
• 通讯作者: Katsuhiro Moriya

Classification of totally real and totally geodesic submanifolds of comnpact 5-symmetric spaces

紧五对称空间全实数和全测地线子流形的分类

• 发表时间: 2006
• 期刊: Journal of Mathematical Society of Japan 58・1
• 作者: Kokubu;Masatoshi; Rossman;Wayne; Umehara;Masaaki; Yamada;Kotaro;Mitsuhiro Itoh;Masatoshi Kokubu;Hiroyuki Tasaki;Katsuhiro Moriya;Katsuya Mashimo;Koji Tojo
• 通讯作者: Koji Tojo