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Alexey V. Bolsinov

Publications and source records attributed to Alexey V. Bolsinov.

At least 19 recordsLinked to original sources
arXiv2026

Local description of gl-regular Haantjes operators

We study Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. Our main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex; the latter case was ignored in many previous papers on this and related topics.

math.DG↗
arXiv2026

Duality of operator Frobenius algebras and solution of Eisenhart-Stäckel problem in the non-diagonal case

We study Frobenius algebras of operator fields and introduce a novel notion of duality for them. We show that, under the assumption that the operator fields forming the Frobenius algebra are mutual symmetries, the operator fields in the dual Frobenius algebra are also mutual symmetries. This result allows one to construct new infinite-dimensional integrable systems of hydrodynamic type starting from a given one. As the main application, we solve the long-standing Eisenhart--Stäckel problem for any Segre characteristic and in arbitrary dimension: namely, we describe all nondegenerate finite-dimensional integrable systems whose integrals are quadratic in momenta such that the corresponding $(1,1)$-tensors commute as operator fields.

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arXiv2025

Stäckel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals

We construct integrable Hamiltonian systems such that functionally independent Poisson commuting integrals are quadratic in the momenta. Unlike the classical Stäckel setting, we allow the associated self-adjoint $(1,1)$-tensors $K_α$ to be non-diagonalisable and have Jordan blocks and points where the Segre characteristic changes. Our construction is covariant and is based on Nijenhuis geometry: starting from a gl-regular Nijenhuis operator $L$ and its symmetry algebra, we obtain a large class of such integrable systems in a coordinate-free and signature-independent way; it is explicit once we have chosen a gl-regular Nijnhuis operator. In the diagonalisable case, our construction reproduces the Stäckel construction, and in dimension $n=2$ it recovers all known systems of this type; for $n\ge 3$ most of our systems are new. Finally, we establish applications to infinite-dimensional integrable systems of hydrodynamic type: namely, we show that for Killing $(1,1)$-tensors $ K_α$ corresponding to our example the evolutionarly PDE system of hydrodynamic type $u_t = K_α(u)u_x$ is integrable. We describe its symmetries, and use generalised reciprocal transformations to reduce it to a system with constant coefficient matrices.

nlin.SI↗
arXiv2025

On the Jordan-Chevalley decomposition problem for operator fields in small dimensions and Tempesta-Tondo conjecture

We explore the Jordan-Chevalley decomposition problem for an operator field in small dimensions. In dimensions three and four, we find tensorial conditions for an operator field $L$, similar to a nilpotent Jordan block, to possess local coordinates in which $L$ takes a strictly upper triangular form. We prove the Tempesta-Tondo conjecture for higher order brackets of Frölicher-Nijenhuis type.

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arXiv2025

Integrability of the magnetic geodesic flow on the sphere with a constant 2-form

We prove a recent conjecture of Dragovic et al arXiv2504.20515 stating that the magnetic geodesic flow on the standard sphere $S^n\subset \mathbb R^{n+1}$ whose magnetic 2-form is the restriction of a constant 2-form from $\mathbb{R}^{n+1}$ is Liouville integrable. The integrals are quadratic and linear in momenta.

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arXiv2024

Research problems on relations between Nijenhuis geometry and integrable systems

The paper surveys open problems and questions related to interplay between the theory of integrable systems with infinitely and finitely many degrees of freedom and Nijenhuis geometry. This text has grown out from preparatory materials for the series of research symposia and workshops on Nijenhuis geometry and integrable systems held at SMRI (Sydney) and MATRIX (Creswick) in February 2022 and at La Trobe University (Melbourne) and MATRIX in February 2024, and from the open problem sessions at these events. It includes both relatively simple questions to get familiar with the topic, as well as challenging problems that are of great importance for the field.

math.DG↗
arXiv2024

Finite-dimensional reductions and finite-gap type solutions of multicomponent integrable PDEs

The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known equations such as the Korteweg--de Vries, coupled KdV, Harry Dym, coupled Harry Dym, Camassa--Holm, multicomponent Camassa--Holm, Dullin--Gottwald--Holm, and Kaup--Boussinesq equations. We suggest a methodology for constructing a series of solutions for all systems in the family. The crux of the approach lies in reducing this system to a dispersionless integrable system which is a special case of linearly degenerate quasilinear systems actively explored since the 1990s and recently studied in the framework of Nijenhuis geometry. These infinite-dimensional integrable systems are closely connected to certain explicit finite-dimensional integrable systems. We provide a link between solutions of our multicomponent PDE systems and solutions of this finite-dimensional system, and use it to construct animations of multi-component analogous of soliton and cnoidal solutions.

math-ph↗
arXiv2023

Orthogonal separation of variables for spaces of constant curvature

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and the Stäckel matrices.

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arXiv2023

Applications of Nijenhuis Geometry V: geodesically equivalent metrics and finite-dimensional reductions of certain integrable quasilinear systems

We describe all metrics geodesically compatible with a gl-regular Nijenhuis operator $L$. The set of such metrics is large enough so that a generic local curve $γ$ is a geodesic for a suitable metric $g$ from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from $L$ preserves the property of $γ$ to be a $g$-geodesic. This implies that every metric $g$ geodesically compatible with $L$ gives us a finite dimensional reduction of this PDE system. We show that its restriction onto the set of $g$-geodesics is naturally equivalent to the Poisson action of $\mathbb{R}^n$ on the cotangent bundle generated by the integrals coming from geodesic compatibility.

math.DG↗
arXiv2023

Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures

The paper contains two lines of results: the first one is a study of symmetries and conservation laws of gl-regular Nijenhuis operators. We prove the splitting Theorem for symmetries and conservation laws of Nijenhuis operators, show that the space of symmetries of a gl-regular Nijenhuis operator forms a commutative algebra with respect to (pointwise) matrix multiplication. Moreover, all the elements of this algebra are strong symmetries of each other. We establish a natural relationship between symmetries and conservation laws of a gl-regular Nijenhuis operator and systems of the first and second companion coordinates. Moreover, we show that the space of conservation laws is naturally related to the space of symmetries in the sense that any conservation laws can be obtained from a single conservation law by multiplication with an appropriate symmetry. In particular, we provide an explicit description of all symmetries and conservation laws for gl-regular operators at algebraically generic points. The second line of results contains an application of the theoretical part to a certain system of partial differential equations of hydrodynamic type, which was previously studied by different authors, but mainly in the diagonalisable case. We show that this system is integrable in quadratures, i.e., its solutions can be found for almost all initial curves by integrating closed 1-forms and solving some systems of functional equations. The system is not diagonalisable in general, and construction and integration of such systems is an actively studied and explicitly stated problem in the literature.

math.DG↗
arXiv2022

Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures

We consider multicomponent local Poisson structures of the form $\mathcal P_3 + \mathcal P_1$, under the assumption that the third order term $\mathcal P_3$ is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. of linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular we show that each such Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with edges labeled by numbers $λ_α$'s and vertices labeled by natural numbers whose sum is the dimension of the manifold. These pencils are naturally related to certain (polynomial, in the most nondegenerate case) pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the Nijenhuis pencil.

math.DG↗
arXiv2022

Applications of Nijenhuis Geometry IV: multicomponent KdV and Camassa-Holm equations

We construct a new series of multicomponent integrable PDE systems that contain as particular example (with appropriately chosen parameters) many famous integrable systems including KdV, coupled KdV, Harry Dym, coupled Harry Dym, Camassa-Holm, multicomponent Camassa-Holm, Kaup-Boussinesq systems. The series contains also integrable systems with no low-component analogues.

nlin.SI↗
arXiv2020

Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is $(n+1)(n+2)/2$ dimensional; we describe it completely and show that it is maximal. Another has dimension $\le n+2$ and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension $n+2$ is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

math.DG↗
arXiv2019

Nijenhuis Geometry

This work is the first, and main, of the series of papers in progress dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that should be understood in much wider context than before: from local description at generic points to singularities and global analysis. The goal of the present paper is to introduce terminology, develop new important techniques (e.g., analytic functions of Nijenhuis operators, splitting theorem and linearisation), summarise and generalise basic facts (some of which are already known but we give new self-contained proofs), and more importantly, to demonstrate that the research programme proposed in the paper is realistic by proving a series of new, not at all obvious, results.

math.DG↗
arXiv2017

Local description of Bochner-flat (pseudo-)Kähler metrics

The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms of their curvature operators. We also give a local description of weakly Bochner-flat metrics defined by the property that the Bochner tensor has vanishing divergence. Our results are based on the local normal forms for c-projectively equivalent metrics. As a by-product, we also describe all Kähler-Einstein metrics admitting a c-projectively equivalent one.

math.DG↗
arXiv2015

Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics

Two Kähler metrics on a complex manifold are called c-projectively equivalent if their $J$-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kähler metrics of arbitrary signature and use this description to prove the classical Yano-Obata conjecture: we show that on a closed connected Kähler manifold of arbitrary signature, any c-projective vector field is an affine vector field unless the manifold is $CP^n$ with (a multiple of) the Fubini-Study metric. As a by-product, we prove the projective Lichnerowicz conjecture for metrics of Lorentzian signature: we show that on a closed connected Lorentzian manifold, any projective vector field is an affine vector field.

math.DG↗