Affine hypersurfaces and superintegrable systems

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 hypersurface normalisations in $R^{n+1}$ ($n⩾2$). We also formulate the necessary and sufficient conditions non-degenerate relative affine hypersurface normalisations in $R^{n+1}$ 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).