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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2018-01-31
Accepted: 2018-03-26
Published online: 2018-04-10

DOI: 10.1016/j.crma.2018.03.012

Author's affiliations:

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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