Nijenhuis Geometry and Integrable Systems — Problem 8

Definitions and mathematical statement

Let $M$ be a smooth real manifold of dimension $n$, and let $L\colon TM\to TM$ be a smooth field of endomorphisms. Its Nijenhuis torsion is \[ N_L(X,Y)=[LX,LY]-L[LX,Y]-L[X,LY]+L^2[X,Y] \] for smooth vector fields $X,Y$. The operator $L$ is Nijenhuis if $N_L=0$. It is $\mathrm{gl}$-regular if, at every point $p$, the minimal polynomial of $L_p$ has degree $n$. In dimension two this means that $L_p$ is not a scalar multiple of the identity at any point.

A standard interpretation of a metric geodesically compatible with $L$ is a smooth nondegenerate symmetric tensor $g$ such that $L$ is self-adjoint with respect to $g$ and \[ (\nabla_XL)Y =\frac12\left({\,\mathrm{d}}(\operatorname{tr}L)(Y)X +g(X,Y)\operatorname{grad}_g(\operatorname{tr}L)\right), \] where $\nabla$ is the Levi--Civita connection of $g$, and $g(\operatorname{grad}_g h,X)={\,\mathrm{d}} h(X)$. This convention is provided to make a possible interpretation explicit; the supplied excerpt does not define geodesic compatibility or specify the allowed metric signature, and its intended convention requires verification.

Requested construction, Problem 8. For each $\mathrm{gl}$-regular Nijenhuis operator $L$ in dimension two, construct explicitly a geodesically compatible metric. The source does not specify whether the construction is local near every point or global on the given manifold. Consequently, a global existence assertion, a choice of signature, and any additional invertibility requirement on $L$ are not imposed here. The real smooth setting above is a conventional interpretation rather than an explicit hypothesis supplied by the excerpt.

Additional requested generalization. In arbitrary coordinates $x^1,\ldots,x^n$, assume that a conservation law is supplied as an explicit function $f(x^1,\ldots,x^n)$. Generalize the formula referred to in the source as the above-mentioned formula from [nijappl5] to these coordinates. The supplied record contains neither that formula nor the bibliographic identification of [nijappl5], and it does not define the conservation-law condition on $f$. This part therefore specifies a requested generalization but does not determine a precise formula or a complete set of hypotheses. Its relationship to the dimension-two construction also remains unspecified.

Short English statement

For every two-dimensional $\mathrm{gl}$-regular Nijenhuis operator, explicitly construct a geodesically compatible metric. The record also asks for a coordinate generalization of an omitted formula using a given conservation-law function. Metric signature, local versus global scope, and the missing formula and conservation-law definition require clarification from the source. The supplied registry labels the problem open; this status is not independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $801155$, exact source problem code catalogue entry, title Nijenhuis Geometry and Integrable Systems --- Problem 8. The supplied record attributes the problem to Bolsinov, Konyaev, and Matveev, with proposed year 2024, and labels its status open.
• Bolsinov, Konyaev, Matveev, Research Problems on Nijenhuis Geometry and Integrable Systems (2024), Problem 8. Supplied source URL: https://arxiv.org/abs/2410.04276. The provenance records extraction as arxiv_latex and access on 2026-10-16. These are supplied provenance statements; no independent source visit or literature verification is claimed.
• The statement refers to [nijappl5], but the supplied record includes neither its bibliographic entry nor the referenced formula.