Bundles of Weyl structures and invariant calculus for parabolic
geometries

D 2023

Bundles of Weyl structures and invariant calculus for parabolic geometries

ČAP, Andreas and Jan SLOVÁK

Basic information

Original name

Bundles of Weyl structures and invariant calculus for parabolic geometries

Authors

ČAP, Andreas (40 Austria) and Jan SLOVÁK (203 Czech Republic, belonging to the institution)

Edition

Rhode Island (USA), The Diverse World of PDEs : Geometry and Mathematical Physics, p. 53-72, 20 pp. 2023

Publisher

American Mathematical Society

Other information

Language

English

Type of outcome

Stať ve sborníku

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

Publication form

printed version "print"

References:

URL

RIV identification code

RIV/00216224:14310/23:00134301

Organization unit

Faculty of Science

ISBN

978-1-4704-7147-7

ISSN

DOI

http://dx.doi.org/10.1090/conm/788/15819

Keywords in English

Cartan geometry; parabolic geometry; Weyl structures; connections; symmetry; differential operator

Abstract

V originále

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 $\Upsilon$.

Links

GX19-28628X, research and development project

Name: Homotopické a homologické metody a nástroje úzce související s matematickou fyzikou

Investor: Czech Science Foundation

Citovat

ČAP, Andreas and Jan SLOVÁK. Bundles of Weyl structures and invariant calculus for parabolic geometries. In Krasil′shchik I. S., Sossinsky A. B., Verbovetsky A. M. The Diverse World of PDEs : Geometry and Mathematical Physics. Rhode Island (USA): American Mathematical Society, 2023, p. 53-72. ISBN 978-1-4704-7147-7. Available from: https://dx.doi.org/10.1090/conm/788/15819.

@inproceedings{2342381,
   author = {Čap, Andreas and Slovák, Jan},
   address = {Rhode Island (USA)},
   booktitle = {The Diverse World of PDEs : Geometry and Mathematical Physics},
   doi = {http://dx.doi.org/10.1090/conm/788/15819},
   editor = {Krasil′shchik I. S., Sossinsky A. B., Verbovetsky A. M.},
   keywords = {Cartan geometry; parabolic geometry; Weyl structures; connections; symmetry; differential operator},
   howpublished = {tištěná verze "print"},
   language = {eng},
   location = {Rhode Island (USA)},
   isbn = {978-1-4704-7147-7},
   pages = {53-72},
   publisher = {American Mathematical Society},
   title = {Bundles of Weyl structures and invariant calculus for parabolic geometries},
   url = {https://bookstore.ams.org/view?ProductCode=CONM/788},
   year = {2023}
}

TY  - JOUR
ID  - 2342381
AU  - Čap, Andreas - Slovák, Jan
PY  - 2023
TI  - Bundles of Weyl structures and invariant calculus for parabolic geometries
PB  - American Mathematical Society
CY  - Rhode Island (USA)
SN  - 9781470471477
KW  - Cartan geometry
KW  - parabolic geometry
KW  - Weyl structures
KW  - connections
KW  - symmetry
KW  - differential operator
UR  - https://bookstore.ams.org/view?ProductCode=CONM/788
N2  - 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 $\Upsilon$.
ER  -

ČAP, Andreas and Jan SLOVÁK. Bundles of Weyl structures and invariant calculus for parabolic geometries. In Krasil′shchik I. S., Sossinsky A. B., Verbovetsky A. M. \textit{The Diverse World of PDEs : Geometry and Mathematical Physics}. Rhode Island (USA): American Mathematical Society, 2023, p.~53-72. ISBN~978-1-4704-7147-7. Available from: https://dx.doi.org/10.1090/conm/788/15819.

Detailed Information on Publication Record