On dual-projectively equivalent connections associated to second order superintegrable systems
Andreas Vollmer
TL;DR
This work studies torsion-free affine connections that arise from irreducible second order superintegrable Hamiltonian systems and shows they are dual-projectively equivalent. By organizing the data into non-degenerate and generalized semi-degenerate (n+2-parameter and n+1-parameter) families, the authors construct induced connections and associated tensors, then apply Ivanov's dual-projective criterion to prove equivalences among the natural connections, including those governing information geometry and restricted-potential spaces. A key outcome is that the induced connections, their g-compatible dual, and the semi-degenerate induced connection are all dual-projectively equivalent under precise symmetry/compatibility conditions, with Weylian structure interpretations illuminating the geometry. The results unify several geometric strands—affine differential geometry, Weyl manifolds, and the integrability of second order superintegrable systems—and suggest broader applicability to abundant systems and affine-hypersurface theory.
Abstract
Pre-geodesics of an affine connection are the curves that are geodesics after a reparametrization (the analogous concept in Kähler geometry is known as J-planar curves). Similarly, dual-geodesics on a Riemannian manifold are curves along which the 1-forms associated to the velocity are preserved after a reparametrization. Superintegrable systems are Hamiltonian systems with a large number of independent constants of the motion. They are said to be second order if the constants of the motion can be chosen to be quadratic polynomials in the momenta. Famous examples include the Kepler-Coulomb system and the isotropic harmonic oscillator. We show that certain torsion-free affine connections which are naturally associated to certain second order superintegrable systems share the same dual-geodesics.