arXiv subjects

Alexey Bolsinov

Publications and source records attributed to Alexey Bolsinov.

13 recordsLinked to original sources

arXiv2026

Beltrami problem in dimension two: local normal forms

Two (pseudo-)Riemannian metrics are said to be geodesically equivalent if they share the same geodesics considered as unparametrized curves. In 1865, E. Beltrami posed the problem of describing all geodesically equivalent metric pairs locally. At generic points, this problem was solved by Dini in the Riemannian setting and later extended to the pseudo-Riemannian case by Bolsinov, Matveev, and Pucacco. In the present paper, we solve the Beltrami problem in dimension two by providing a complete local classification of geodesically equivalent pseudo-Riemannian metrics in a neighbourhood of a singular point. This yields a full solution of the problem in dimension two.

math.DG↗

arXiv2025

On Separation of Variables for Symmetric Spaces of Rank1

We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu, 2020, by showing that a symmetric space of rank 1 has diagonal coordinates if and only if it has constant sectional curvature. This implies that orthogonal separation of variables on a symmetric space of rank 1 is possible only in the constant sectional curvature case. We show that on the complex projective space $\mathbb{C}P^n$ and on complex hyperbolic space $\mathbb{C}H^n$, with $n\ge 2$, separating coordinates necessarily have precisely $n$ ignorable coordinates. In view of results of Boyer et al, 1983 and 1985, and later results of Winternitz et al, 1994, this completes the description of separation of variables on $\mathbb{C}P^n$ for all $n$ and on $\mathbb{C}H^n$ for $n=2,3$.

math.DG↗

arXiv2020

Nijnehuis Geometry III: gl-regular Nijenhuis operators

We study Nijenhuis operators, that is, (1,1)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. We prove the existence of a coordinate system in which the operator takes first or second companion form, and give a local describtion of such operators. We apply this local description to study singular points. In particular, we obtain their normal forms in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces. This paper is an important step in the research programme suggested in arXiv:1903.04603 and arXiv:1903.06411.

math.DG↗

arXiv2019

Jordan-Kronecker invariants of Lie algebra representations and degrees of invariant polynomials

For an arbitrary representation $ρ$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $ρ$. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of $ρ$. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

math.RT↗

arXiv2018

Smooth invariants of focus-focus singularities and obstructions to product decomposition

We study focus-focus singularities (also known as nodal singularities, or pinched tori) of Lagrangian fibrations on symplectic $4$-manifolds. We show that, in contrast to elliptic and hyperbolic singularities, there exist homeomorphic focus-focus singularities which are not diffeomorphic. Furthermore, we obtain an algebraic description of the moduli space of focus-focus singularities up to smooth equivalence, and show that for double pinched tori this space is one-dimensional. Finally, we apply our construction to disprove Zung's conjecture which says that any non-degenerate singularity can be smoothly decomposed into an almost direct product of standard singularities.

math.SG↗

arXiv2018

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems with two degrees of freedom

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.

math.SG↗

arXiv2016

Argument shift method and sectional operators: applications to differential geometry

This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems on Lie algebras and a rather different area of mathematics studying projectively equivalent Riemannian and pseudo-Riemannian metrics.

math.DG↗

arXiv2014

Some remarks about Mishchenko-Fomenko subalgebras

We discuss and compare two different approaches to the notion of Mishchenko--Fomenko subalgebras in Poisson-Lie algebras of finite-dimensional Lie algebras. One of them, commonly accepted by the algebraic community, uses polynomial $\Ad^*$-invariants. The other is based on formal $\Ad^*$-invariants and allows one to deal with arbitrary Lie algebras, not necessarily algebraic. In this sense, the latter is more universal.

math.RT↗

arXiv2014

On one class of holonomy groups in pseudo-Riemannian geometry

We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, we prove the following theorem. Let g be a nondegenerate bilinear form on a vector space V, and L:V -> V a g-symmetric operator. Then the identity component of the centraliser of L in SO(g) is a holonomy group for a suitable Levi-Civita connection.

math.DG↗

arXiv2013

Singularities of bi-Hamiltonian systems

We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types.

math-ph↗

arXiv2012

Jordan-Kronecker invariants of finite-dimensional Lie algebras

For any finite-dimensional Lie algebra we introduce the notion of Jordan-Kronecker invariants, study their properties and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko-Fomenko's argument shift method.

math.RT↗

arXiv2012

Complete commutative subalgebras in polynomial Poisson algebras: a proof of the Mischenko--Fomenko conjecture

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra $\goth g$ there exists a complete set of commuting polynomials on its dual space $\goth g^*$. In terms of the theory of integrable Hamiltonian systems this means that the dual space $\goth g^*$ endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. Recently this conjecture has been proved by S.T. Sadetov. Following his idea, we give an explicit geometric construction for commuting polynomials on $\goth g^*$ and consider some examples.

math.DG↗

arXiv1996

Topology of energy surfaces and existence of transversal Poincaré sections

Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincaré section. We show that there are topological obstacles for its existence such that only in the cases of $S^1\times S^2$ and $T^3$ such a Poincaré section can exist.

chao-dyn↗