Nijenhuis Geometry and Integrable Systems — Problem 9

Definitions and mathematical statement

The supplied record contains two requests whose full formulation depends on omitted source context. Let $x=(x^1,\ldots,x^n)$ denote an arbitrary local coordinate system, and write $f=f(x^1,\ldots,x^n)$ for the explicitly given function that the source assumes to be a conservation law. The excerpt does not specify the system for which it is a conservation law, its defining equation, its codomain, or the required regularity.

For an endomorphism $L_p$ of an $n$-dimensional tangent space, the standard meaning of $\mathrm{gl}$-regularity is \[ \dim\{A\in\operatorname{End}(T_pM):AL_p=L_pA\}=n, \] equivalently, that the minimal polynomial of $L_p$ has degree $n$. This standard definition clarifies the terminology only; the excerpt does not supply the ambient category, scalar field, or full hypotheses on the operator field $L$.

1. Generalise the previously mentioned formula attributed in the source to [nijappl5] to arbitrary coordinates $x^1,\ldots,x^n$, assuming that a conservation law is supplied explicitly as $f(x^1,\ldots,x^n)$. The referenced formula, its inputs and outputs, and its hypotheses are absent from the supplied record, so the identity or construction to be generalised cannot be stated explicitly.
2. Construct new explicit examples, in arbitrary dimension, of metrics geodesically compatible with $\mathrm{gl}$-regular operators, including examples at singular points. The supplied excerpt does not define geodesic compatibility or singular points, specify the admissible dimensions or metric signatures, or state the additional hypotheses on the operators. These conditions must be recovered from the source before a complete system of equations and quantifiers can be given.

The second request appears after the displayed Problem environment in the supplied record and is retained as an additional requested construction. The record labels the problem open; this is a supplied classification, not an independently verified mathematical assessment.

Short English statement

Extend the source's referenced formula to arbitrary coordinates using an explicitly known conservation law, and construct explicit geodesically compatible metrics for $\mathrm{gl}$-regular operators at singular points in arbitrary dimension. The omitted formula and missing geometric definitions prevent a fully specified mathematical statement.

Sources

• Local UnsolvedMath registry, permanent ID $801156$, exact source code catalogue entry, title Nijenhuis Geometry and Integrable Systems --- Problem 9. 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 9. Supplied source URL: https://arxiv.org/abs/2410.04276. The provenance reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source inspection.
• The statement references [nijappl5], but the supplied record includes neither its bibliographic entry nor the formula attributed to it. No author, title, or URL for this reference can be recovered from the supplied evidence.