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arXiv · 2503.11852

Singularities of two-dimensional Nijenhuis operators

Dinmukhammed Akpan
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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.

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Dinmukhammed Akpan. 2025-03-14. Singularities of two-dimensional Nijenhuis operators. https://doi.org/10.1007/s40879-022-00585-x

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