Affine hypersurfaces and superintegrable systems
Vicente Cortés, Andreas Vollmer
TL;DR
This work builds a precise bridge between abundant second-order conformally superintegrable systems and affine hypersurface geometry by establishing a one-to-one correspondence between abundant manifolds $(M,g,S,t)$ (n>=3) and abundant hypersurface normalisations (and their co-normals) in $\mathbb{R}^{n+1}$. The core idea is that the abundant structure induces a relative cubic $C$ and metric $G$ whose dual connection realises the ambient affine geometry, with the key relation $dt=3u$ tying the conformal scale to the cubic trace. The authors prove existence and uniqueness of the immersion realizing the correspondence, show compatibility with conformal rescalings, and extend the construction to dimension $2$ with a specialized framework. Applications include encoding isotropic harmonic oscillators and constant-curvature systems as abundant hypersurfaces, and identifying graph normalisations as a natural realisation in their framework. Overall, the results provide a robust geometric method to construct and classify abundant systems via affine differential geometry, unifying conformal superintegrability with Blaschke-type immersions.
Abstract
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).