  In this work, we investigate the effects of a nontrivial topology (and
geometry) of a system considering \textit{interacting} and
\textit{noninteracting} particle modes, which are restricted to follow a closed
path over the torus surface. In order to present a prominent thermodynamical
investigation of this system configuration, we carry out a detailed analysis
using statistical mechanics within the grand canonical ensemble approach to
deal with \textit{noninteracting} fermions. In an analytical manner, we study
the following thermodynamic functions in such context: the Helmholtz free
energy, the mean energy, the magnetization and the susceptibility. Further, we
take into account the behavior of Fermi energy of the thermodynamic system.
Finally, we briefly outline how to proceed in case of \textit{interacting}
fermions.
