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 (). We also formulate the necessary and sufficient conditions non‐degenerate relative affine hypersurface normalisations in 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).