Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian $\mathcal{T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians ${\mathcal{T}}_r$ ($r=a,b,c,d$) in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential ${\mathcal{U}}_r$ leading to ${\mathcal{H}}_r={\mathcal{T}}_r+{\mathcal{U}}_r$. Secondly, we study the superintegrability of the four Hamiltonians ${\tilde{\mathcal{H}}}_r={\mathcal{H}}_r\mathrm{/}{\mu }_r$, where ${\mu }_r$ is a certain position-dependent mass, that enjoys the same separability as the original system ${\mathcal{H}}_r$. All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.