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Vladimir S. Matveev

Publications and source records attributed to Vladimir S. Matveev.

At least 19 recordsLinked to original sources
arXiv2026

Sophus Lie's problem on two-dimensional metrics with projective symmetries: completing the local classification

We complete the local classification, up to isometry, of $2$-dimensional pseudo-Riemannian metrics (i.e. both Riemannian and Lorentzian), admitting a projective symmetry algebra of dimension at least two. The new contribution is the treatment of the non-regular case: we obtain a complete list of mutually non-isometric normal forms in neighbourhoods of points where the action of the projective Lie symmetry algebra fails to be regular, including all singular behaviours that can occur. Together with the known classification in the regular case, this completes the solution of the problem posed by Sophus Lie in 1882.

math.DG↗
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.

math.DG↗
arXiv2025

Lax pairs for BKM hierarchy

We construct Lax pairs for the recently (2023) introduced integrable PDE systems known as the BKM equations. As many known and previously studied integrable systems are special cases of the BKM systems, our construction provides Lax pairs for many integrable hierarchies, including previously studied ones such as Camassa-Holm, Dullin-Gottwald-Holm, cKdV, Ito, and Marvan-Pavlov, as well as new ones. The corresponding pair is related to a Sturm-Liouville operator on the real line whose potential depends rationally on the spectral parameter.

nlin.SI↗
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

Variationality of conformal geodesics in dimension 3

Conformal geodesics form an invariantly defined family of unparametrized curves in a conformal manifold generalizing unparametrized geodesics/paths of projective connections. The equation describing them is of third order, and it was an open problem whether they are given by an Euler--Lagrange equation. In dimension 3 (the simplest, but most important from the viewpoint of physical applications) we demonstrate that the equation for unparametrized conformal geodesics is variational.

math.DG↗
arXiv2025

Thoughts about potentials with finite-band spectrum and finite-dimensional reductions of integrable systems

We repeat, using methods developed for BKM systems, the famous results of S. Novikov (1974), J. Moser (1981, 1982) , and A. Veselov (1980) that relate Schrödinger-Hill operators with finite-band spectra, solutions of the Neumann system, and certain solutions of the KdV equations. Our general motivation is to determine whether it is possible to apply inverse scattering methods to BKM systems, and in the conclusion, we indicate initial observations in this direction.

math-ph↗
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.

math.DG↗
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.

math.DG↗
arXiv2025

Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures

We study two-dimensional Riemannian metrics which are superintegrable in the class of polynomial in momenta integrals. The study is based on our main technical result, Theorem 3, which states that the Poisson bracket of two polynomial in momenta integrals is an algebraic function of the integrals and of the Hamiltonian. We conjecture that two-dimensional superintegrable Riemannian metrics are necessary real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. Small modification of the arguments, discussed in the paper, provides a methods to construct new superintegrable systems. We prove a special case of the above conjecture which is sufficient to show that the metrics constructed by K. Kiyohara in 2001, which admit irreducible polynomial in momenta integrals of arbitrary high degree $k$, are not superintegrable and in particular do not admit nontrivial polynomial in momenta integral of degree less than $k$. This result solves Conjectures (b) and (c) explicitly formulated in Bolsinov, KOzlov and Fomenko in 1995.

nlin.SI↗
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↗
arXiv2024

Killing tensors on reducible spaces

We prove that on the product of two Riemannian manifolds one of which is compact, any Killing tensor is reducible, that is, is the sum of products of Killing tensors on the factors. The same is true for the lifts to the universal cover of Killing tensors on a compact manifold with reducible holonomy. We give a local description of Killing tensors on product manifolds and present an example of a complete product manifold whose factors are locally irreducible which admits an irreducible Killing tensor field.

math.DG↗
arXiv2024

If a Minkowski billiard is projective, it is the standard billiard

In the recent paper arXiv:2405.13258, the first author of this note proved that if a billiard in a convex domain in $\mathbb{R}^n$ is simultaneously projective and Minkowski, then it is the standard Euclidean billiard in an appropriate Euclidean structure. The proof was quite complicated and required high smoothness. Here we present a direct simple proof of this result which works in $C^1$-smoothness. In addition we prove the semi-local and local versions of the result

math.DS↗
arXiv2024

Integrable geodesic flows with simultaneously diagonalisable quadratic integrals

We show that if $n$ functionally independent commutative quadratic in momenta integrals for the geodesic flow of a Riemannian or pseudo-Riemannian metric on an $n$-dimensional manifold are simultaneously diagonalisable at the tangent space to every point, then they come from the Stäckel construction, so the metric admits orthogonal separation of variables.

math.DG↗
arXiv2023

Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields

Every Killing tensor field on the space of constant curvature and on the complex projective space can be decomposed into the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial in the velocities integral of the geodesic flow is a polynomial in the linear integrals). This fact led to the natural question on whether this property is shared by Killing tensor fields on all Riemannian symmetric spaces. We answer this question in the negative by constructing explicit examples of quadratic Killing tensor fields which are not quadratic forms in the Killing vector fields on the quaternionic projective spaces $\mathbb{H} P^n, n \ge 3$, and on the Cayley projective plane $\mathbb{O} P^2$.

math.DG↗