Nijenhuis geometry. III: \(\mathrm{gl}\)-regular Nijenhuis operators (Q6118799)

Summary: We study Nijenhuis operators, that is, \((1,1)\)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are \(\mathrm{gl}\)-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 \(\mathrm{gl}\)-regular Nijenhuis operators near singular points in dimension two and discover topological restrictions for the existence of \(\mathrm{gl}\)-regular Nijenhuis operators on closed surfaces.