A microscopic theory of superconductivity in metallic quantum rings can be derived by taking into account the effects of double-connectedness on the electron properties of the nanoring, within the framework of the Bardeen-Cooper-Schrieffer (BCS) theory. To this aim, an analytical quantum confinement model is presented which allows one to describe the topology of Fermi sea for a 3D nanostructureNanostructure. After introducing the essential concepts on the example of thin films or nanosheets, where a topological transition in the Fermi surface is shown to arise at a critical thickness due to quantum confinement, the same approach is subsequently applied to the nanoring geometry. It is found that the electronic density of states undergoes two topological transitions upon reducing the geometric parameters of the ring, separating three distinct regimes. In each regime, the Fermi energy displays a peculiar dependence on the geometric parameters of the ring. These analytical closed-form results are then implemented in the BCS theoryBCS theory to yield predictions of the superconducting critical temperature \(T_c\) as a function of the nanoring geometric parameter values.