Singularities of two-dimensional Nijenhuis operators

Abstract

A Nijenhuis operator $L$ is a $(1,1)$-tensor field on a smooth manifold $M$ with vanishing Nijenhuis torsion ${\mathcal{N}}_L$. At each point $x\in M$, the algebraic type of $L(x)$ is characterized by its Jordan normal form. In this paper, we study singularities of a two-dimensional Nijenhuis operator in the case when its trace has a non-zero differential at the singular point. A description of such singularities reduces to studying the smoothness of some function, which is a fraction depending on partial derivatives of the determinant of $L$. We completely describe singularities for some special classes of functions. We also obtained interesting examples of Nijenhuis operators and their singularities.

References in corpus (3)

• Nijenhuis Geometry
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Nijenhuis geometry II: Left-symmetric algebras and linearization problem for Nijenhuis operators

Cited by in corpus (2)

• Almost Differentially Nondegenerate Nijenhuis Operators
• Elementary Differential Singularities of Three-Dimensional Nijenhuis Operators