On the Lichnerowicz conjecture for CR manifolds with mixed signature

Jeffrey S. Case; Sean N. Curry; Vladimir S. Matveev

doi:10.1016/j.crma.2018.03.012

Geometry/Differential geometry

On the Lichnerowicz conjecture for CR manifolds with mixed signature
[Sur la conjecture de Lichnerowicz pour les variétés CR de signature mixte]

Présenté par : the Editorial Board

Jeffrey S. Case ¹ ; Sean N. Curry ² ; Vladimir S. Matveev ³

¹ 109 McAllister Building, Penn State University, University Park, PA 16802, USA
² Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA
³ Institut für Mathematik, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 532-537.

Résumé
Abstract

Nous construisons des exemples de variétés CR non dégénérées avec une forme de Levi de signature $(p, q)$ , $2 \leq p \leq q$ , qui sont compactes, non localement CR plates et admettent des champs de vecteurs CR essentiels. Nous construisons également un exemple d'une variété CR non dégénérée et non compacte, de signature $(1, n - 1)$ , qui n'est pas localement CR plate et admet un champ de vecteurs CR essentiel. Ceci fournit des contre-exemples aux analogues de la conjecture de Lichnerowicz pour les variétés CR de signature mixte.

We construct examples of nondegenerate CR manifolds with Levi form of signature $(p, q)$ , $2 \leq p \leq q$ , which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature $(1, n - 1)$ that is not locally CR flat and admits an essential CR vector field. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.

Reçu le : 2018-01-31
Accepté le : 2018-03-26
Publié le : 2018-04-10

DOI : 10.1016/j.crma.2018.03.012

Affiliations des auteurs :

Jeffrey S. Case ¹ ; Sean N. Curry ² ; Vladimir S. Matveev ³

¹ 109 McAllister Building, Penn State University, University Park, PA 16802, USA
² Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA
³ Institut für Mathematik, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

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     author = {Jeffrey S. Case and Sean N. Curry and Vladimir S. Matveev},
     title = {On the {Lichnerowicz} conjecture for {CR} manifolds with mixed signature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {532--537},
     publisher = {Elsevier},
     volume = {356},
     number = {5},
     year = {2018},
     doi = {10.1016/j.crma.2018.03.012},
     language = {en},
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Jeffrey S. Case; Sean N. Curry; Vladimir S. Matveev. On the Lichnerowicz conjecture for CR manifolds with mixed signature. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 532-537. doi : 10.1016/j.crma.2018.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.03.012/

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Jacob W. Erickson Higher Rank Parabolic Geometries with Essential Automorphisms and Nonvanishing Curvature, Transformation Groups, Volume 30 (2025) no. 1, p. 203 | DOI:10.1007/s00031-022-09780-2

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