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

在均匀空间中积分几何形状的研究中，线性同位素作用的轨道上的整合起着关键作用。许多重要的例子是严峻的子曼群。代表Ikawa和Sakai将Riemann对称对的线性同位素作用的严重轨道归类为球体中的线性同位素作用。当详细检查这些严峻的轨道时，许多严峻的轨道不仅具有第二种基本形式的对称性，而且具有某种全球对称性。因为这种全局对称性是削弱反射式子手术的定义的特征，所以它被命名为弱反思性的submanifold并检查了其基本属性。弱反射性的子手势是严重的子曼叶夫，而严峻的亚体则是最小的亚体。除了上述严重轨道的分类外，弱反射轨道还可以分类。结果表明，具有退化高斯图的轨道变成了一个弱反射的子人物……更多的iFold，并且已经概括了这一点，尽管结果是带有脱离高斯映射的同一性1的轨道以前变得更加严峻。也就是说，具有变性高斯图的轨道是弱反射性的，反射性轨道弱的轨道是朴素的。对Riemann Symmetry对线性同位素作用的轨道的研究对于领导各种关系的关系非常重要。实际上，代表引入的多个Kaehler角度的概念可以自然地从轨道的几何学观点中引起，并且当配制了复杂空间形式的运动学公式时，从轨道中的几何考虑角色中获得的角色起着基本作用。此外，多个kaehler角及其基本特性是整体几何形状重建的重要基础，这使得过去已经发展的价值概念，几年是基础。轨道几何形状的观点对于在未来方向上研究积分几何形状是必不可少的。较少的

项目成果

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

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

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

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

Transferred kinematic formulae in two point homogeneous spaces

两点齐次空间中的传递运动学公式

• 发表时间: 2007
• 期刊: Tsukuba J. Math 31
• 作者: 浅子和美;加納悟;倉澤資成;T.Sakai
• 通讯作者: T.Sakai