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Singularities of two-dimensional Nijenhuis operators

Dinmukhammed Akpan

TL;DR

This work analyzes singularities of two-dimensional Nijenhuis operators under the condition ${d\,\mathrm{tr}\,L \neq 0}$ by reducing the reconstruction of ${L}$ to the smoothness of a fraction tied to ${\det L}$. Central to the analysis is the discriminant ${g(x,y)=\frac{\mathrm{tr}^2 L}{4}-\det L}$, which classifies admissible singularities and yields explicit local models (notably ${L_1^{\pm}}$ and ${L_2^{\pm}}$) for nondegenerate cases with ${g_y\neq0}$, as well as higher-order (cubic) degeneracies. The paper provides a complete description of admissible discriminants, including the special cases ${g=(\frac{x}{2}-\alpha)^2}$ and ${g\equiv0}$, and demonstrates a range of concrete examples (e.g., ${g(x,y)=G(y)}$ and homogeneous polynomials) that illustrate the construction and linearizability of the resulting operators. These results connect singularity types to the geometry of the characteristic map and contribute to the broader understanding of integrable structures in Nijenhuis geometry.

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.

Singularities of two-dimensional Nijenhuis operators

TL;DR

This work analyzes singularities of two-dimensional Nijenhuis operators under the condition by reducing the reconstruction of to the smoothness of a fraction tied to . Central to the analysis is the discriminant , which classifies admissible singularities and yields explicit local models (notably and ) for nondegenerate cases with , as well as higher-order (cubic) degeneracies. The paper provides a complete description of admissible discriminants, including the special cases and , and demonstrates a range of concrete examples (e.g., and homogeneous polynomials) that illustrate the construction and linearizability of the resulting operators. These results connect singularity types to the geometry of the characteristic map and contribute to the broader understanding of integrable structures in Nijenhuis geometry.

Abstract

A Nijenhuis operator is a -tensor field on a smooth manifold with vanishing Nijenhuis torsion . At each point , the algebraic type of 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 . We completely describe singularities for some special classes of functions. We also obtained interesting examples of Nijenhuis operators and their singularities.

Paper Structure

This paper contains 4 sections, 8 theorems, 24 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The main definitions
  3. Singularities of Nijenhuis operators
  4. Examples of Nijenhuis operators

Key Result

Theorem 3.1

Let $L$ be a two-dimensional Nijenhuis operator and $\mathop{\mathrm{tr}}\nolimits L = x$, $\det L = f(x,y)$, with $f_y(x,y)\neq 0$. Then the point $(0,0)$ is differentially non-degenerate and $L$ is given in coordinates $(x,y)$ by the following formula:

Figures (5)

  • Figure 1: The level lines of the real eigenvalues of the Niejenhus operator $L_1^{+}$ from Theorem \ref{['T3']}.
  • Figure 2: The level lines of the real eigenvalues of the Niejenhus operator $L_1^{-}$ from Theorem \ref{['T3']}. The domain with complex eigenvalues is given by the inequality $x^2 - 4y^2 < 0$.
  • Figure 3: The level lines of the real eigenvalues of the Niejenhus operator $L_2^{+}$ from Theorem \ref{['T3']}. For $L_2^{-}$ all eigenvalues are complex.
  • Figure 4: The level lines of the real eigenvalues of the Niejenhus operator $L_1$ from Theorem \ref{['T4']}.The domain with complex eigenvalues is given by the inequality $y < 0$.
  • Figure 5: The level lines of the real eigenvalues of the Niejenhus operator $L_2$ from Theorem \ref{['T4']}. The domain with complex eigenvalues is given by the inequality $x^2 + 16y^3 < 0$.

Theorems & Definitions (31)

  • Definition 2.1: bib8
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: bib1
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1: bib6
  • Theorem 3.2
  • Remark 3.3
  • proof
  • ...and 21 more