Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties

Definitions and mathematical statement

Let $S$ be a closed real analytic surface, meaning a compact real analytic $2$-manifold without boundary. A real analytic $(1,1)$-tensor field $L$ is a real analytic bundle endomorphism $L:TS\to TS$. Its Nijenhuis torsion is the tensor \[ N_L(X,Y)=[LX,LY]-L[LX,Y]-L[X,LY]+L^2[X,Y] \] for vector fields $X,Y$. It is a Nijenhuis operator if $N_L=0$.

Construction problem. Construct real analytic examples $(S,L)$ with $N_L=0$ such that every $L_x:T_xS\to T_xS$ has real eigenvalues and those eigenvalues are distinct on an open dense subset of $S$. Equivalently, writing \[ D(x)=(\operatorname{tr}L_x)^2-4\det L_x, \] the required eigenvalue conditions are \[ D(x)\geq0\quad\text{for all }x\in S, \qquad \{x\in S:D(x)>0\}\text{ is dense in }S. \] The latter set is automatically open. Repeated eigenvalues are therefore allowed on the exceptional closed set. The source requests examples, without prescribing a genus, orientability, or a construction on every closed surface.

The supplied research summary marks the problem solved and reports such constructions by Bolsinov--Konyaev--Matveev, citing arXiv:2007.09506 and publication in Revista Matemática Iberoamericana (2024). This is retained as the record's literature claim rather than an independently checked proof.

Short English statement

Construct a real analytic Nijenhuis endomorphism of the tangent bundle of a closed surface whose eigenvalues are real everywhere and distinct on an open dense set. The registry reports examples by Bolsinov--Konyaev--Matveev; no topology of the surface is prescribed in the question.

Sources

• Local UnsolvedMath registry, permanent ID 2100510, source code catalogue entry, "Open Problems in Integrable Systems --- Nijenhuis operators: singular points and global properties"; supplied status solved, classification SOLVED-IN-LITERATURE, updated 2026-08-06.
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 5.10, https://arxiv.org/abs/1804.03737. The record reports source-TeX extraction and access on 2026-07-29. Its background says the source presents an open item and current status requires release review. Rights: NEEDS_REVIEW; public arXiv source TeX inspected according to the record.
• Bolsinov--Konyaev--Matveev, https://arxiv.org/abs/2007.09506, reportedly published in Revista Matemática Iberoamericana (2024), as supplied in the research summary. No article title, pages, or theorem number are provided.