  For a particle that is constrained on an ($N-1$)-dimensional ($N\geq2$)
curved surface, the Cartesian components of its momentum in $N$-dimensional
flat space is believed to offer a proper form of momentum for the particle on
the surface, which is called the geometric momentum as it depends on the mean
curvature. Once the momentum is made general covariance, the spin connection
part can be interpreted as a gauge potential. The present study consists in two
parts, the first is a discussion of the general framework for the general
covariant geometric momentum. The second is devoted to a study of a Dirac
fermion on a two-dimensional sphere and we show that there is the generalized
total angular momentum whose three cartesian components form the $su(2)$
algebra, obtained before by consideration of dynamics of the particle, and we
demonstrate that there is no curvature-induced geometric potential for the
fermion.
