Nijnehuis Geometry III: gl-regular Nijenhuis operators

TL;DR

This paper investigates gl-regular Nijenhuis operators, establishing local coordinate forms, classifying singular points, and revealing topological restrictions on their existence on closed surfaces.

Contribution

It provides a local description and normal forms for gl-regular Nijenhuis operators, advancing understanding of their structure and topological constraints.

Findings

Existence of coordinate systems with companion form for these operators

Normal forms for singular points in dimension two

Topological restrictions on closed surfaces for such operators

Abstract

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 describtion of such operators. We apply this local description to study singular points. In particular, we obtain their normal forms in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces. This paper is an important step in the research programme suggested in arXiv:1903.04603 and arXiv:1903.06411.