arXiv subjects

Andreas Vollmer

Publications and source records attributed to Andreas Vollmer.

At least 19 recordsLinked to original sources

arXiv2026

Second-order superintegrable systems from semi-simple and nilpotent Frobenius structures

Recently, it was shown that a rich class of second-order (maximally) superintegrable systems has an underpinning Hesse-Frobenius structure, i.e. a Frobenius structure that is compatible with a Hessian structure such that the Hessian pre-potential is also a Frobenius pre-potential. Hence, these superintegrable systems arise, locally, from (possibly non-unital) Frobenius algebras. We present a conification to lift systems of non-zero constant sectional curvature to flat ones, thus reducing the problem on the Frobenius algebra side to the associative case. Furthermore, we employ a direct product construction to generate higher-dimensional second-order maximally superintegrable systems on flat spaces. We apply this approach to some basic semi-simple and nilpotent algebras and explicitly construct the arising second-order superintegrable systems using a Neumann series expansion. We find that all non-degenerate second-order maximally superintegrable systems in three dimensions arise from these examples.

math-ph↗

arXiv2026

Affine hypersurfaces and superintegrable systems

It was recently shown that under mild assumptions second-order conformally superintegrable systems can be encoded in a $(0,3)$-tensor, called structure tensor. For abundant systems, this approach led to algebraic integrability conditions that essentially allow one to restore a system from the knowledge of its structure tensor in a point on the manifold. Here we study the geometric structure formalising such systems, which we call an abundant manifold. The underlying Riemannian manifold is necessarily conformally flat. We establish a correspondence between these superintegrable systems and the geometry of affine hypersurfaces. More precisely, we show that abundant manifolds correspond to certain non-degenerate relative affine hypersurfaces normalisations in $\mathbb R^{n+1}$ ($n\ge 2$). We also formulate the necessary and sufficient conditions non-degenerate relative affine hypersurface normalisations in $\mathbb R^{n+1}$ need to satisfy, if they arise from abundant manifolds. These relative affine hypersurface normalisations are called abundant hypersurface normalisations. Both for abundant manifolds and for relative affine hypersurface normalisations a natural concept of conformal equivalence can be defined. We prove that they are compatible, permitting us to identify conformal classes of abundant manifolds with abundant hypersurface immersions (without specified normalisation).

math.DG↗

arXiv2026

Torsion and semi-degeneracy of second-order maximally superintegrable systems

The isotropic harmonic oscillator and the Kepler-Coulomb system are pivotal models in the Sciences. They are two examples of second-order (maximally) superintegrable (Hamiltonian) systems. These systems are classified in dimension two. A partial classification exists in dimension three. In this paper, our focus is on second-order superintegrable systems with a $(n+1)$-parameter potential with $n\geq3$. We find that these systems are underpinned by an information-geometric structure, namely the structure of a statistical manifold with torsion. We obtain a necessary and sufficient condition for such systems to extend to non-degenerate systems, i.e. to admit a maximal family of compatible potentials. The condition is geometric: we show that a $(n+1)$-parameter potential is the restriction of a non-degenerate potential if and only if a certain trace-free tensor field vanishes. We interpret this condition as the requirement that a certain affine connection has vectorial torsion. We also show that the condition for a system to be extendable is conformally invariant, allowing us to extend our results to second-order conformally superintegrable systems with a $(n+1)$-parameter potential.

math.DG↗

arXiv2025

Hessian geometry and Frobenius manifolds with curvature

A Riemannian metric is called Hessian if, locally, it can be written as the Hessian of a function called the Hessian potential. A (flat) Manin-Frobenius manifold is a flat Riemannian manifold furnished with a commutative and associative product compatible with the metric, such that a certain potentiality property is satisfied. Curved Frobenius manifolds generalize this concept to spaces with non-vanishing curvature, and they have applications in supersymmetric mechanics and within the theory of submanifolds. Curved Frobenius manifolds naturally arise from Hessian metrics, and we find that they, conceptually, are the typical non-trivial examples. We obtain that Curved Frobenius structures on constant curvature spaces are consistent with a Hessian structure, if they satisfy a closed prolongation system of finite type. Consistency means that the Frobenius potential and the Hessian potential can be identified. As an application, we show that certain second-order maximally superintegrable systems correspond 1-to-1 to Curved Frobenius structures that are consistent with a Hessian structure.

math.DG↗

arXiv2025

Second-order superintegrable systems and Weylian geometry

Abundant second-order maximally conformally superintegrable Hamiltonian systems are re-examined, revealing their underlying natural Weyl structure and offering a clearer geometric context for the study of Stäckel transformations (also known as coupling constant metamorphosis). This also allows us to naturally extend the concept of conformal superintegrability from the realm of conformal geometries to that of Weyl structures. It enables us to interpret superintegrable systems of the above type as semi-Weyl structures, a concept related to statistical manifolds and affine hypersurface theory.

math.DG↗

arXiv2025

On dual-projectively equivalent connections associated to second order superintegrable systems

Pre-geodesics of an affine connection are the curves that are geodesics after a reparametrization (the analogous concept in Kähler geometry is known as J-planar curves). Similarly, dual-geodesics on a Riemannian manifold are curves along which the 1-forms associated to the velocity are preserved after a reparametrization. Superintegrable systems are Hamiltonian systems with a large number of independent constants of the motion. They are said to be second order if the constants of the motion can be chosen to be quadratic polynomials in the momenta. Famous examples include the Kepler-Coulomb system and the isotropic harmonic oscillator. We show that certain torsion-free affine connections which are naturally associated to certain second order superintegrable systems share the same dual-geodesics.

math.DG↗

arXiv2025

Manifolds with a commutative and associative product structure that encodes superintegrable Hamiltonian systems

We show that two natural and a priori unrelated structures encapsulate the same data, namely certain commutative and associative product structures and a class of superintegrable Hamiltonian systems. More precisely, consider a Euclidean space of dimension at least three, equipped with a commutative and associative product structure that satisfies the conditions of a Manin-Frobenius manifold, plus one additional compatibility condition. We prove that such a product structure encapsulates precisely the conditions of a so-called abundant structure. Such a structure provides the data needed to construct a family of second-order (maximally) superintegrable Hamiltonian systems of second order. We prove that all abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise in this way. As an example, we present the Smorodinski-Winternitz Hamiltonian system.

math.DG↗

arXiv2024

On Haantjes tensors for second-order superintegrable systems

The vanishing of the Haantjes tensor is an important property that has been linked, for instance, to the existence of separation coordinates and the integrability of systems of hydrodynamic type. We discuss the vanishing of the Haantjes tensor for operator fields that admit a large number of so-called conservation laws. In particular, we investigate Haantjes-zero Killing tensor fields that are associated with second-order superintegrable systems.

math-ph↗

arXiv2024

The c-projective symmetry algebras of Kähler surfaces

Let $M$ be a Kähler manifold with complex structure $J$ and Kähler metric $g$. A c-projective vector field is a vector field on $M$ whose flow sends $J$-planar curves to $J$-planar curves, where $J$-planar curves are analogs of what (unparametrised) geodesics are for pseudo-Riemannian manifolds (without complex structure). The c-projective symmetry algebras of Kähler surfaces with essential (i.e., non-affine) c-projective vector fields are computed.

math.DG↗

arXiv2024

Abundant Superintegrable Systems and Hessian Structures

We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant superintegrable systems on Riemannian manifolds of constant sectional curvature fall into this class. We explicitly compute the natural Hessian coordinates for examples of non-degenerate second-order superintegrable systems in dimensions two and three.

nlin.SI↗

arXiv2024

Superintegrable systems on conformal surfaces

We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of dimension three and higher. The governing equations of non-degenerate second order superintegrability in dimension two are structurally significantly different from those valid in higher dimensions. Specifically, we find conformally covariant structural equations, allowing one to classify the (conformal classes of) non-degenerate second order superintegrable systems on conformal surfaces geometrically. We then specialise to second order properly superintegrable systems on surfaces with a (pseudo-)Riemannian metric and obtain structural equations in accordance with the known equations for Euclidean space. We finally give a single explicit set of purely algebraic equations defining the variety parametrising such systems on all constant curvature surfaces.

math.DG↗

arXiv2024

Torsion-free connections of second-order maximally superintegrable systems

Second-order (maximally) conformally superintegrable systems play an important role as models of mechanical systems, including systems such as the Kepler-Coulomb system and the isotropic harmonic oscillator. The present paper is dedicated to understanding non- and semi-degenerate systems. We obtain "projective flatness" results for two torsion-free connections naturally associated to such systems. This viewpoint sheds some light onto the interrelationship of properly and conformally (second-order maximally) superintegrable systems from a geometrical perspective. It is shown that the semi-degenerate secondary structure tensor can be viewed as the Ricci curvature of a natural torsion-free connection defined by the primary structure tensor (and similarly in the non-degenerate case). It is also shown that properly semi-degenerate systems are characterised, similar to the non-degenerate case, by the vanishing of the secondary structure tensor.

math.DG↗

arXiv2022

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

We show that the definition of a second order superintegrable system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant notion of superintegrability. Conformal equivalence is the natural extension of the well-known Stäckel transform, which in turn originates from the classical Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.

math.DG↗

arXiv2021

3-dimensional Levi-Civita metrics with projective vector fields

Projective vector fields are the infinitesimal transformations whose local flow preserves geodesics up to reparametrisation. In 1882 Sophus Lie posed the problem of describing 2-dimensional metrics admitting a non-trivial projective vector field, which was solved in recent years. In the present paper, we solve the analog of Lie's problem in dimension 3, for Riemannian metrics and, more generally, for Levi-Civita metrics of arbitrary signature.

math.DG↗

arXiv2021

Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics

A non-degenerate second-order maximally conformally superintegrable system in dimension 2 naturally gives rise to a quadric with position dependent coefficients. It is shown how the system's Stäckel class can be obtained from this associated quadric.The Stäckel class of a second-order maximally conformally superintegrable system is its equivalence class under Stäckel transformations, i.e., under coupling-constant metamorphosis.

nlin.SI↗

arXiv2020

An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem - based on a proof that the classification space for irreducible non-degenerate second-order superintegrable systems is naturally endowed with the structure of a quasi-projective variety with a linear isometry action. On constant curvature manifolds our approach leads to a single, simple and explicit algebraic equation defining the variety classifying superintegrable Hamiltonians that satisfy all relevant integrability conditions generically. In particular, this includes all non-degenerate superintegrable systems known to date and shows that our approach is manageable in arbitrary dimension. Our work establishes the foundations for a complete classification of second-order superintegrable systems in arbitrary dimension, derived from the geometry of the classification space, with many potential applications to related structures such as quadratic symmetry algebras and special functions.

math.DG↗

arXiv2019

Normal forms of two-dimensional metrics admitting exactly one essential projective vector field

We give a complete list of mutually non-diffeomorphic normal forms for the two-dimensional metrics that admit one essential (i.e., non-homothetic) projective vector field. This revises a result from the literature and extends the results of two papers, by R.L. Bryant & G. Manno & V.S. Matveev (2008) and V.S. Matveev (2012) respectively, solving a problem posed by Sophus Lie in 1882.

math.DG↗

arXiv2019

Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be considered the projective analog of (conformal) Stäckel equivalence (coupling constant metamorphosis). Then, we discuss the transformation behavior for projectively equivalent superintegrable systems and find that the potential on a projectively equivalent geometry can be reconstructed from a characteristic vector field. Moreover, potentials of projectively equivalent Hamiltonians follow a linear superimposition rule. The techniques are applied to several examples. In particular, we use them to classify, up to Stäckel equivalence, the superintegrable systems on geometries with one, non-trivial projective symmetry.

math.DG↗