We study the existence of global smooth solutions for the quasi-linear wave equation by an internal local damping when initial data are close to a given equilibrium. Our interest is in studying the structure of the damping region, which guarantees the existence of global solutions. Our results show that the structure of the damping region depends on the geometric properties of a Riemannian metric, based on the coefficients and the equilibrium of the system. Some geometrical conditions are presented to obtain the damping region.