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For a free particle that non-relativistically moves on a curved surface,there are curvature-induced quantum potentials that significantly influence thesurface quantum states, but the experimental results in topological insulators,whenever curved or not, indicate no evidence of such a potential, implying thatthere does not exist such a quantum potential for the relativistic particles,constrained on the surface or not. Within the framework of Dirac quantizationscheme, we demonstrate a general result that for a Dirac fermion on atwo-dimensional curved surface of revolution, no curvature-induced quantumpotential is permissible.

No existence of the geometric potential for a Dirac fermion on two-dimensional curved surfaces of revolution