  We consider the tilted-hexatic Hamiltonian on the fluctuating membranes. A
renormalization-group analysis leads us to find two critical regions; one
corresponds to the strong coupling regime of the gradient cross coupling, the
other to the weak coupling regime. In the strong coupling regime, we find the
locked tilted-hexatic to liquid phase transition when disclinations and
vortices unbind simultaneously. On the other hand, in the weak coupling regime
we find four phases; the unlocked tilted-hexatic phase, the hexatic phase, the
tilted phase, and liquid phase. Disclinations and vortices unbind
independently. The crinkled-to-crumpled transition of the fluctuating
tilted-hexatic membranes is also described. The crinkled phase in the strong
coupling regime is stiffer than that in the weak coupling regime.
