Weyl结构丛与抛物几何的不变微积分
Bundles of Weyl structures and invariant calculus for parabolic geometries
AI总结：
针对抛物几何中Weyl联络的仿射不变量构造问题，本文提出一种通用微积分，给出仅张量依赖于形变$Υ$的所有仿射不变量的自然构造程序，为抛物几何微分不变量研究提供关键第一步。
摘要：
一百多年来，人们发展了多种概念来理解共形黎曼几何与射影几何中的几何对象场以及对象之间的不变微分算子。近年来，针对整类抛物几何（即以齐性空间$G/P$为模型的Cartan几何，其中$P$是半单李群$G$中的抛物子群），研究者提出了若干通用工具。与共形黎曼结构和射影结构类似，所有这些几何都确定了一类特异仿射联络，这类联络具有以微分1-形式$Υ$为模型的仿射结构。它们对应于$P$到其约化Levi因子的约化，并且仿照共形情形被称为Weyl结构。在该框架下，微分不变量的标准定义是这些联络的仿射不变量，即不依赖于该类中具体选取的不变量。本文中，我们描述了一种通用微积分，它为确定这类不变量提供了重要的第一步。我们提出了一种自然程序，用于构造Weyl联络的所有仿射不变量，这些不变量仅以张量方式依赖于形变$Υ$。
英文摘要：
For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian and projective geometries. More recently, several general tools were presented for the entire class of parabolic geometries, i.e., the Cartan geometries modelled on homogeneous spaces $G/P$ with $P$ a parabolic subgroup in a semi-simple Lie group $G$. Similarly to conformal Riemannian and projective structures, all these geometries determine a class of distinguished affine connections, which carry an affine structure modelled on differential 1-forms $Υ$. They correspond to reductions of $P$ to its reductive Levi factor, and they are called the Weyl structures similarly to the conformal case. The standard definition of differential invariants in this setting is as affine invariants of these connections, which do not depend on the choice within the class. In this article, we describe a universal calculus which provides an important first step to determine such invariants. We present a natural procedure how to construct all affine invariants of Weyl connections, which depend only tensorially on the deformations $Υ$.
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