  We study the properties of a two-dimensional non-relativistic electron gas
(TDEG) constrained on wires along a M\"{o}bius strip. We considered wires
around the strip and along the transverse direction, across the width of the
strip. For each direction, we investigate how the curvature modifies the
electronic states and their corresponding energy spectrum. At the center of the
strip, the wires around the surface form quantum rings whose spectrum depends
on the strip radius $a$. For wires at the edge of the strip, the inner edge
turns into the outer edge. Accordingly, the curvature yields localized states
in the middle of the wire. Along the strip width, the effective potential
exhibits a parity symmetry breaking leading to the localization of the bound
state on one side of the strip.
