Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 062, 18 pages      arXiv:2208.12690

Separation of Variables and Superintegrability on Riemannian Coverings

Claudia Maria Chanu a and Giovanni Rastelli b
a) Dipartimento di Scienze Umane e Sociali, Università della Valle d'Aosta, Italy
b) Dipartimento di Matematica, Università di Torino, Italy

Received January 11, 2023, in final form August 23, 2023; Published online September 03, 2023

We introduce Stäckel separable coordinates on the covering manifolds $M_k$, where $k$ is a rational parameter, of certain constant-curvature Riemannian manifolds with the structure of warped manifold. These covering manifolds appear implicitly in literature as connected with superintegrable systems with polynomial in the momenta first integrals of arbitrarily high degree, such as the Tremblay-Turbiner-Winternitz system. We study here for the first time multiseparability and superintegrability of natural Hamiltonian systems on these manifolds and see how these properties depend on the parameter $k$.

Key words: Riemannian coverings; integrable systems; separable coordinates.

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