Bundles of Weyl structures and invariant calculus for parabolic geometries

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Authors

ČAP Andreas SLOVÁK Jan

Year of publication 2023
Type Article in Proceedings
Conference The Diverse World of PDEs : Geometry and Mathematical Physics
MU Faculty or unit

Faculty of Science

Citation
Web https://bookstore.ams.org/view?ProductCode=CONM/788
Doi http://dx.doi.org/10.1090/conm/788/15819
Keywords Cartan geometry; parabolic geometry; Weyl structures; connections; symmetry; differential operator
Description 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 $\Upsilon$. 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 $\Upsilon$.
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