Nijenhuis Geometry and Integrable Systems — Problem 13

Definitions and mathematical statement

For terminology, let $g$ be a nondegenerate metric on a smooth manifold $M$, and let $\nabla$ be its Levi--Civita connection. A symmetric covariant tensor $K$ of rank $r\geq 1$ is a Killing tensor if \[ \nabla_{(i_0}K_{i_1\dots i_r)}=0, \] where parentheses denote symmetrization. This definition supplies terminology only: the record does not specify the intended rank, metric signature, or geometric category.

A Frobenius manifold, in the usual terminology, carries a flat nondegenerate metric and a commutative, associative multiplication on tangent spaces with a parallel unit, such that the metric is invariant under multiplication and the covariant derivative of the tensor $c(X,Y,Z)=g(X\circ Y,Z)$ is totally symmetric. The record does not specify whether additional conventions, such as an Euler vector field, are required.

First question (the supplied Problem 13). Is there a construction from a Frobenius manifold of a Killing tensor of the kind associated in the source with geometric Poisson brackets of order $3$? The source describes these as Killing tensors whose covariant derivative satisfies an additional relation, but does not supply that relation. It also says that certain Frobenius manifolds have already been used to construct such tensors. Consequently, the requested novelty, the admissible class of Frobenius manifolds, and the precise conditions on the output tensor remain unresolved; the question cannot be made into a fully specified existence statement from this record alone.

Second question (appended in the supplied record). Do there exist two distinct compatible Poisson structures of order $3$, at least one of which is not Darboux--Poisson?

Here compatibility means that the two brackets $\{\cdot,\cdot\}_1$ and $\{\cdot,\cdot\}_2$ are defined on a common space of observables and every scalar linear combination \[ \{F,G\}_{a,b}=a\{F,G\}_1+b\{F,G\}_2 \] is again a Poisson bracket. A Poisson bracket is bilinear and skew-symmetric, satisfies the Leibniz rule, and obeys \[ \{F,\{G,H\}\}+\{G,\{H,F\}\}+\{H,\{F,G\}\}=0. \] The common space, scalar field, meaning of differential order $3$, and definition of Darboux--Poisson are not given. The record also does not say whether "different" means unequal structures or inequivalent structures under a specified class of transformations. These qualifications must be fixed before the second question has a precise geometric formulation.

Short English statement

The record asks for a construction of the relevant Killing tensors from Frobenius manifolds, and also asks whether two distinct third-order Poisson structures can be compatible when at least one is not Darboux--Poisson. Essential geometric conditions and the intended scope of the construction are missing. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801160$, exact source code catalogue entry, title "Nijenhuis Geometry and Integrable Systems --- Problem 13." The record attributes the problem to Bolsinov, Konyaev, and Matveev, with proposed year $2024$, and supplies the status label "open."
• Bolsinov, Konyaev, and Matveev, Research Problems on Nijenhuis Geometry and Integrable Systems (2024), Problem 13, as attributed by the supplied record. Supplied source URL: https://arxiv.org/abs/2410.04276.
• Supplied provenance: extraction method arxiv_latex; source access date 2026-10-16; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The access date and attribution are supplied provenance, not a claim of independent source consultation. The appended compatibility question is present in the record, but its precise placement in the cited source is not identified.