  An infinite crystal can be constructed by an infinite number of parallel
two-dimensional (hkl) crystal planes coupled to each other. For crystals with
negligible spin-orbit coupling, we report a rigorous proof of a criterion on
the non-existence of surface states in a semi-infinite crystal with the crystal
symmetry. The forward transfer to be the same as the backward one, called as
F-B dynamical symmetry, is key to realize the criterion. Based on lattice model
Hamiltonian with coupling between the nearest neighbor crystal planes only, we
prove that a cut crystal will not be able to accommodate any surface states if
the original infinite crystal has reflection symmetry about every crystal plane
which results in F-B symmetry. The criterion provide a platform to simply
conclude whether surface states exist or not in a cut crystal. For any such
crystals, the non-existence or existence of surface states depends on the cut
direction of the crystal plane. Since the spin-orbit coupling breaks the chiral
symmetry, resulting in the F-B asymmetry, surface states can emerge in the
(hkl) cut crystal with spin-orbit coupling.
