Nijenhuis geometry III: gl-regular Nijenhuis operators

  • Alexey V. Bolsinov

    Loughborough University, UK
  • Andrey Yu. Konyaev

    Moscow State University, and Moscow Center for Fundamental and Applied Mathematics, Russia
  • Vladimir S. Matveev

    Friedrich Schiller Universität Jena, Germany
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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.

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Alexey V. Bolsinov, Andrey Yu. Konyaev, Vladimir S. Matveev, Nijenhuis geometry III: gl-regular Nijenhuis operators. Rev. Mat. Iberoam. 40 (2024), no. 1, pp. 155–188

DOI 10.4171/RMI/1416