arXiv subjects

Vladimir Matveev

Publications and source records attributed to Vladimir Matveev.

9 recordsLinked to original sources

arXiv2026

Quadratic Killing tensors on classical Lie groups are decomposable

A Killing tensor field on a Riemannian manifold $(M,g)$ is a covariant symmetric tensor field whose contraction with the velocity vector along a geodesic produces a homogeneous polynomial first integral of the geodesic flow. Such a tensor is called \emph{decomposable} if it lies in the subalgebra generated by Killing vector fields; equivalently, the corresponding polynomial integral is then a polynomial in the linear integrals coming from infinitesimal isometries. On spaces of constant sectional curvature and on the complex projective space, every Killing tensor field is decomposable. By contrast, the quaternionic projective spaces and the Cayley projective plane admit indecomposable quadratic Killing tensor fields. We prove that every quadratic Killing tensor field on the compact classical Lie groups $\mathrm{SO}(n)$, $\mathrm{Spin}(n)$, $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$, equipped with a bi-invariant Riemannian metric, is decomposable; equivalently, every quadratic first integral of the geodesic flow on these groups is a quadratic polynomial in the linear first integrals.

math.DG↗

arXiv2026

On Killing tensors on Riemannian symmetric spaces

A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A Killing tensor field is called decomposable if it is a polynomial in Killing vector fields. In this paper, we first prove that the study of Killing tensor fields on symmetric spaces can be reduced to the case of compact irreducible ones. Then we introduce the class of top slot Killing tensor fields. We obtain an explicit and elegant description of such tensor fields and prove that the quadratic Killing tensor fields are spanned by the top-slot ones. We also show that quadratic Killing tensor fields on the quaternionic projective space and on the Cayley projective space are spanned by the indecomposable ones constructed in our earlier paper and the decomposable ones. This completes the classification of quadratic Killing tensor fields on Riemannian symmetric spaces of rank one.

math.DG↗

arXiv2025

Self-adjoint quantization of St\"ackel integrable systems

We show that quadratic Hamiltonians in involution coming from a St\"ackel system are quantizable, in the sense that one can construct commutative self-adjoint operators whose symbols are the quadratic Hamiltonians. Moreover, they allow multiplicative separation of variables. This proves a conjecture explicitly formulated in [3].

math.DG↗

arXiv2021

Strange Hadron Spectroscopy with Secondary KL Beam in Hall D

We propose to create a secondary beam of neutral kaons in Hall D at Jefferson Lab to be used with the GlueX experimental setup for strange hadron spectroscopy. The superior CEBAF electron beam will enable a flux on the order of $1\times 10^4~K_L/sec$, which exceeds the flux of that previously attained at SLAC by three orders of magnitude. The use of a deuteron target will provide first measurements ever with neutral kaons on neutrons. The experiment will measure both differential cross sections and self-analyzed polarizations of the produced $Λ$, $Σ$, $Ξ$, and $Ω$ hyperons using the GlueX detector at the Jefferson Lab Hall D. The measurements will span CM $\cosθ$ from $-0.95$ to 0.95 in the range W = 1490 MeV to 2500 MeV. The new data will significantly constrain the partial wave analyses and reduce model-dependent uncertainties in the extraction of the properties and pole positions of the strange hyperon resonances, and establish the orbitally excited multiplets in the spectra of the $Ξ$ and $Ω$ hyperons. Comparison with the corresponding multiplets in the spectra of the charm and bottom hyperons will provide insight into he accuracy of QCD-based calculations over a large range of masses. The proposed facility will have a defining impact in the strange meson sector through measurements of the final state $Kπ$ system up to 2 GeV invariant mass. This will allow the determination of pole positions and widths of all relevant $K^\ast(Kπ)$ $S$-,$P$-,$D$-,$F$-, and $G$-wave resonances, settle the question of the existence or nonexistence of scalar meson $κ/K_0^\ast(700)$ and improve the constrains on their pole parameters. Subsequently improving our knowledge of the low-lying scalar nonet in general.

nucl-ex↗

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↗

arXiv2020

Some geometric correspondences for homothetic navigation

In this paper, we provide conceptional explanations for the geodesic and Jacobi field correspondences for homothetic navigation, and then let them guide us to the shortcuts to some well known flag curvature and S-curvature formulas. They also help us directly see the local correspondence between isoparametric functions or isoparametric hypersurfaces, which generalizes the classification works of Q. He and her coworkers for isoparametric hypersurfaces in Randers space forms and Funk spaces.

math.DG↗

arXiv2015

Submaximally symmetric c-projective structures

C-projective structures are analogues of projective structures in the complex setting. The maximal dimension of the Lie algebra of c-projective symmetries of a complex connection on an almost complex manifold of C-dimension $n>1$ is classically known to be $2n^2+4n$. We prove that the submaximal dimension is equal to $2n^2-2n+4+2δ_{3,n}$. If the complex connection is minimal (encoded as a normal parabolic geometry), the harmonic curvature of the c-projective structure has three components and we specify the submaximal symmetry dimensions and the corresponding geometric models for each of these three pure curvature types. If the connection is non-minimal, we introduce a modified normalization condition on the parabolic geometry and use this to resolve the symmetry gap problem. We prove that the submaximal symmetry dimension in the class of Levi-Civita connections for pseudo-Kähler metrics is $2n^2-2n+4$, and specializing to the Kähler case, we obtain $2n^2-2n+3$. This resolves the symmetry gap problem for metrizable c-projective structures.

math.DG↗

arXiv2015

Smoothing 3-dimensional polyhedral spaces

We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.

math.DG↗

arXiv2013

Submaximal metric projective and metric affine structures

We prove that the next possible dimension after the maximal $n^2+2n$ for the Lie algebra of local projective symmetries of a metric on a manifold of dimension $n>1$ is $n^2-3n+5$ if the signature is Riemannian or $n=2$, $n^2-3n+6$ if the signature is Lorentzian and $n>2$, and $n^2-3n+8$ elsewise. We also prove that the Lie algebra of local affine symmetries of a metric has the same submaximal dimensions (after the maximal $n^2+n$) unless the signature is Riemannian and $n=3,4$, in which case the submaximal dimension is $n^2-3n+6$.

math.DG↗