Andreas Vollmer (Hamburg Universität)
Abstract: We introduce a geometric framework for second-order (maximally) superintegrable Hamiltonian systems on Riemannian manifolds of any dimension, encoding the system in a symmetric cubic tensor field (work with J. Kress and K. Schöbel). This approach allows one to interpret Stäckel transformations (coupling constant metamorphosis) of superintegrable systems in terms of a conformal geometry (or, more precisely, a Weylian structure).
The general framework will be exemplified for systems with a maximal number of compatible potentials and of linearly independent integrals of motion (“abundant systems”). This class includes the Smorodinski-Winternitz system, for instance. We find a natural correspondence between abundant systems and affine hypersurfaces (work with V. Cortés). On spaces of constant sectional curvature, abundant systems have an underlying Hessian structure (with J. Armstrong), and on flat space they correspond to solutions of the Witten-Dijkgraaf-Verlinde-Verlinde equation, allowing their identification with Manin-Frobenius manifolds.