Elementary Differential Singularities of Three-Dimensional Nijenhuis Operators
Dinmukhammed Akpan, Andrei Oshemkov
TL;DR
The paper classifies three-dimensional Nijenhuis operators with differential singularities by analyzing the rank of the invariant differential map $\Phi_L$. For ${\rm rk}(d\Phi_L)\equiv 1$, it derives three explicit families of characteristic polynomials corresponding to solvable ODE constraints on the invariants, yielding concrete normal forms. For ${\rm rk}(d\Phi_L)=2$, it studies cases where two invariants are independent and the third has a Morse singularity on their joint level set, giving three canonical normal forms depending on which invariant is Morse, and proving gl-regularity and diagonalizability in these cases. The results provide a complete description of differential singularities of three-dimensional Nijenhuis operators and furnish new explicit examples, expanding the understanding of singularity structures in Nijenhuis geometry and their normal forms.
Abstract
In the paper, three-dimensional Nijenhuis operators are studied that have differential singularities, i.e., such points at which the coefficients of the characteristic polynomials are dependent. The case is studied in which the differentials of all invariants of the Nijenhuis operator are proportional, as well as the case when two invariants are functionally independent and the third defines a fold-type singularity. In particular, new examples of three-dimensional Nijenhuis operators with singularities of the specified type are constructed.