Nijenhuis geometry III: gl-regular Nijenhuis operators

• Alexey V. Bolsinov ^[1] ; Andrey Yu. Konyaev ^[2] ; Vladimir S. Matveev ^[3]
  1. [1] Loughborough University

     Loughborough University

     Charnwood District, Reino Unido

     
  2. [2] Moscow State University

     Moscow State University

     Rusia

     
  3. [3] Friedrich Schiller Universität Jena, Germany

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 1, 2024, págs. 155-188
• Idioma: inglés
• DOI: 10.4171/RMI/1416
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• Resumen

  • We study Nijenhuis operators, that is, (1,1)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. We prove the existence of a coordinate system in which the operator takes first or second companion form, and give a local description of such operators. We apply this local description to study singular points. In particular, we obtain normal forms of gl-regular Nijenhuis operators near singular points in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces.