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In this work, we study the properties of an electron constrained on wiresalong the M\"{o}bius strip. We considered wires around the strip and along thetransverse direction, across the width of the strip. For each direction, weinvestigate how the curvature modifies the electronic states and theircorresponding energy spectrum. At the center of the strip, the wires around thesurface form quantum rings whose spectrum depends on the strip radius $a$. Forwires 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 symmetrybreaking leading to the localization of the bound state on one side of thestrip.

Electronic states in quantum wires on the M\"{o}bius strip