Singularities of three-dimensional linear Nijenhuis operators with functionally independent invariants

Abstract

This work is devoted to the study of singularities of Nijenhuis operators – fundamental
objects of Nijenhuis geometry. Although the Nijenhuis tensor was introduced by Albert
Nijenhuis back in 1951, this field received active development relatively recently thanks to
a series of works by A.V. Bolsinov, A.Yu. Konyaev, and V.S. Matveev.
In dimension two, the classification of linear Nijenhuis operators, operators acting on a
linear space, whose components linearly depend on coordinates, is known. There exists an
important one-to-one correspondence between linear Nijenhuis operators and left-symmetric algebras, which makes their classification an equivalent problem.
Despite its apparent simplicity, the problem remains challenging even for small dimensions
and can be solved only under certain additional constraints. This paper investigates threedimensional linear Nijenhuis operators (or, equivalently, three-dimensional left-symmetric algebras) under the condition of functional independence of the characteristic polynomial coefficients. A complete classification of operators with this additional condition was recently obtained, yielding a list of eight operators.
The main objective of this paper is to study the singularities of such operators. A singular
point is defined as a point in any neighbourhood of which the algebraic type of the operator (Jordan normal form) changes. The paper determines singular points for the considered class of Nijenhuis operators and constructs their sets in three-dimensional space.