Manifolds with a commutative and associative product structure that encodes superintegrable Hamiltonian systems
Andreas Vollmer
TL;DR
The paper demonstrates a 1-to-1 correspondence between Manin-Frobenius manifolds satisfying the compatibility $\nabla_Z\hat P(X,Y)=\hat P(Z\star X,Y)$ and abundant structures that underpin maximally second-order superintegrable Hamiltonian systems on flat manifolds of dimension $n\ge 3$. It constructs an abundant structure from a Manin-Frobenius setting via $S=-3\mathring P$ and $dt=-\frac{3}{n+2}\operatorname{tr}(\hat P)$, and conversely shows how every flat abundant structure arises from a commutative, associative product $\star$ defined by $X\star Y=\hat P(X,Y)$ with a carefully chosen $P$, yielding a full equivalence. The Hessian interpretation ties the geometry to a potential function $\phi$ with $g=\nabla^{\hat P}d\phi$ and $P=\nabla^3\phi$, illuminating the structural compatibility. The Smorodinski-Wintenitz system provides an explicit, higher-dimensional example illustrating the correspondence and the role of the unit field. This framework offers a geometric route to construct and classify abundant second-order superintegrable systems via Manin-Frobenius data.
Abstract
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.