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For a particle that is constrained on an ($N-1$)-dimensional ($N\geq2$)curved surface, the Cartesian components of its momentum in $N$-dimensionalflat space is believed to offer a proper form of momentum for the particle onthe surface, which is called the geometric momentum as it depends on the meancurvature. Once the momentum is made general covariance, the spin connectionpart can be interpreted as a gauge potential. The present study consists in twoparts, the first is a discussion of the general framework for the generalcovariant geometric momentum. The second is devoted to a study of a Diracfermion on a two-dimensional sphere and we show that there is the generalizedtotal angular momentum whose three cartesian components form the $su(2)$algebra, obtained before by consideration of dynamics of the particle, and wedemonstrate that there is no curvature-induced geometric potential for thefermion.

General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere