Essential Parabolic Structures and Their Infinitesimal Automorphisms
Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146856 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Essential Parabolic Structures and Their Infinitesimal Automorphisms / J. Alt // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862614035597361152 |
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| author | Alt, J. |
| author_facet | Alt, J. |
| citation_txt | Essential Parabolic Structures and Their Infinitesimal Automorphisms / J. Alt // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As a corollary of the generalized Ferrand-Obata theorem proved by C. Frances, this proves a generalization of the ''Lichnérowicz conjecture'' for conformal Riemannian, strictly pseudo-convex CR, and quaternionic/octonionic contact manifolds in positive-definite signature. For an infinitesimal automorphism with a singularity, we give a generalization of the dictionary introduced by Frances for conformal Killing fields, which characterizes (local) essentiality via the so-called holonomy associated to a singularity of an infinitesimal automorphism.
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| first_indexed | 2025-11-29T09:33:27Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146856 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-29T09:33:27Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Alt, J. 2019-02-11T17:22:22Z 2019-02-11T17:22:22Z 2011 Essential Parabolic Structures and Their Infinitesimal Automorphisms / J. Alt // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53B05; 53C05; 53C17; 53C24 DOI:10.3842/SIGMA.2011.039 https://nasplib.isofts.kiev.ua/handle/123456789/146856 Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As a corollary of the generalized Ferrand-Obata theorem proved by C. Frances, this proves a generalization of the ''Lichnérowicz conjecture'' for conformal Riemannian, strictly pseudo-convex CR, and quaternionic/octonionic contact manifolds in positive-definite signature. For an infinitesimal automorphism with a singularity, we give a generalization of the dictionary introduced by Frances for conformal Killing fields, which characterizes (local) essentiality via the so-called holonomy associated to a singularity of an infinitesimal automorphism. I am grateful to Charles Frances for discussions about the methods used in [8] and [10], and to Felipe Leitner for useful comments on an earlier version of the text. The anonymous referees made many helpful criticisms and suggestions; in particular, I am grateful for the suggestion to reformulate an earlier (equivalent) version of Definition 2.1 in terms of the action of an automorphism on the set of Weyl structures. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Essential Parabolic Structures and Their Infinitesimal Automorphisms Article published earlier |
| spellingShingle | Essential Parabolic Structures and Their Infinitesimal Automorphisms Alt, J. |
| title | Essential Parabolic Structures and Their Infinitesimal Automorphisms |
| title_full | Essential Parabolic Structures and Their Infinitesimal Automorphisms |
| title_fullStr | Essential Parabolic Structures and Their Infinitesimal Automorphisms |
| title_full_unstemmed | Essential Parabolic Structures and Their Infinitesimal Automorphisms |
| title_short | Essential Parabolic Structures and Their Infinitesimal Automorphisms |
| title_sort | essential parabolic structures and their infinitesimal automorphisms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146856 |
| work_keys_str_mv | AT altj essentialparabolicstructuresandtheirinfinitesimalautomorphisms |