论文标题
抛物线几何的Weyl结构和不变计算
Bundles of Weyl structures and invariant calculus for parabolic geometries
论文作者
论文摘要
一百多年来，开发了各种概念，以了解它们之间的几何对象和不变差分运算符的领域，以了解共形的riemannian和投射几何形状。最近，为整个抛物线层的几何形状提供了几种通用工具，即，以同质空间为$ g/p $的cartan几何形状，并以$ p $ p $ pu $ po $ po $ po $ po $ po $ G $ G $ G $ G $ G $ G $。与共形的里曼尼亚和投射结构类似，所有这些几何形状决定了一类杰出的仿射连接，该连接带有以差异1形式$υ$建模的仿射结构。它们对应于将$ p $减少到其还原性李维斯因子，它们被称为与共形案例相似的Weyl结构。在这种情况下，差异不变的标准定义是这些连接的仿射不变的，这不取决于类中的选择。在本文中，我们描述了一个通用的演算，它提供了确定此类不变的重要第一步。我们提出了一个自然的过程，如何构建Weyl连接的所有仿射不变，这仅取决于变形$υ$。
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 $Υ$.
