The Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold without boundary is considered. Let us assume that the dissipative effects are effective in , where is an arbitrary open bounded set with smooth boundary. In the present article we introduce a new class of non compact Riemannian manifolds, namely, manifolds which admit a smooth function f, such that the Hessian of f satisfies the pinching conditions (locally in ), for those ones, there exist a finite number of disjoint open subsets free of dissipative effects such that and for all , , or, in other words, the dissipative effect inside possesses measure arbitrarily small. It is important to be mentioned that if the function f satisfies the pinching conditions everywhere, then it is not necessary to consider dissipative effects inside Omega.