  The Dirac equation in $(2+1)$ dimensions on the toroidal surface is studied
for a massless fermion particle under the action of external fields. Using the
covariant approach based on general relativity, the Dirac operator stemming
from a metric related to the strain tensor is discussed within the
Pseudo-Hermitian operator theory. Furthermore, analytical solutions are
obtained for two cases, namely, constant and position-dependent Fermi velocity.
