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Decompositing of $N+1$-dimensional gradient operator in terms of Gaussiannormal coordinates $(\xi^{0},\xi^{\mu})$, ($\mu=1,2,3,...,N$) and making thecanonical momentum $P_{0}$ along the normal direction $\mathbf{n}$ to behermitian, we obtain $\mathbf{n}P_{0}=-i\hbar\left( \mathbf{n}\partial_{0}-\mathbf{M}_{0}\right) $ with $\mathbf{M}_{0}$ denoting the mean curvaturevector on the surface $\xi^{0}=const.$ The remaining part of the momentumoperator lies on the surface, which is identical to the geometric one.

On relationship between canonical momentum and geometric momentum