Kinematic formulas in Hermitian space forms and applications

埃尔米特空间形式的运动学公式及其应用

• 批准号: 215529441
• 负责人: Professor Dr. Andreas Bernig
• 金额: --
• 依托单位: Schwerpunkt Analysis und Numerik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/215529441?language=en
• 关键词: Kinematic; formulas; Hermitian; space; forms

Hermitian Integral Geometry is the part of integral geometry that concerns Hermitian vector spaces, complex projective and complex hyperbolic spaces, with the actions of their respective isometry groups. A central question is to determine the average number of intersection points of two random geometric subsets in these spaces. In the complex space forms (hermitian vector spaces, complex projective and complex hyperbolic spaces), the corresponding kinematic formulas have been found recently by Bernig-Fu and Bernig-Fu-Solanes, using the algebraic structure on the space of valuations introduced by S. Alesker as well as new algebraic structures on the space of curvature measures. As a result of the computations, the kinematic formulas turn out to be formally identical in these spaces. The first aim of the project is a better understanding of hermitian integral geometry. We will first study the integral geometry of the two-dimensional quaternionic space forms to see whether the kinematic formulas are formally identical, analogous to the complex case. A deeper and more elegant way of stating local kinematic formulas on complex space forms would be by describing the algebra structure (generators and relations) on the dual space to the space of unitarily invariant curvature measures, which is our second aim. The third aim is to describe the multipliers of the Alesker-Fourier transform on smooth translation invariant valuations.

Hermitian积分几何形状是涉及赫米尔族矢量空间，复杂的投影和复杂双曲线空间的整体几何形状的一部分，以及其各自的等轴测组的作用。一个核心问题是确定这些空间中两个随机几何子集的相交点的平均数量。在复杂的空间形式（Hermitian矢量空间，复杂的投影和复杂的双曲线空间）中，最近，Bernig-Fu和Bernig-fu-Solanes最近发现了相应的运动学配方，使用代数结构在S. Alesker以及新代数结构以及在Curval of Curvature space of Curvature space上引入的估值空间上。由于计算，运动公式在这些空间中正式相同。该项目的第一个目的是更好地理解Hermitian积分几何形状。我们将首先研究二维季节空间形式的积分几何形状，以查看运动学公式是否正式相同，类似于复杂情况。在复杂空间形式上说明局部运动学公式的更深层次，更优雅的方式是描述双重空间上的代数结构（发电机和关系）到单位不变的曲率措施的空间，这是我们的第二个目标。第三个目的是描述Alesker-fourier的乘数在平滑翻译不变的估值中的乘数。