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.

埃尔米特积分几何是积分几何的一部分，涉及埃尔米特向量空间、复射影和复双曲空间及其各自等距群的作用。一个中心问题是确定这些空间中两个随机几何子集的平均交点数量。在复空间形式（厄米向量空间、复射影空间和复双曲空间）中，Bernig-Fu 和 Bernig-Fu-Solanes 最近利用 S. Alesker 引入的估价空间上的代数结构以及曲率测度空间上的新代数结构找到了相应的运动学公式。计算结果表明，这些空间中的运动学公式在形式上是相同的。该项目的首要目标是更好地理解厄米积分几何。我们将首先研究二维四元空间形式的积分几何，看看运动学公式是否在形式上相同，类似于复杂的情况。在复杂空间形式上表述局部运动学公式的一种更深入、更优雅的方式是通过将对偶空间上的代数结构（生成元和关系）描述为酉不变曲率测量空间，这是我们的第二个目标。第三个目标是描述平滑平移不变估值上的 Alesker-Fourier 变换的乘数。