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We are concerned with the asymptotic behavior of two different cubic, defocusing and damped nonlinear Schrödinger equations on compact Riemannian manifolds without boundary. Two mechanisms of locally distributed damping are considered: a weak damping and a stronger one. In the first problem, we consider a two-dimensional case and prove that the corresponding energy functional goes to zero as time goes to infinity. The proof is based on a result of propagation of singularities due to Dehman et al. (Math Z 254(4):729-749, 2006) and Strichartz type inequalities due to Burq et al. (Am J Math 126(3):569-605, 2004), combined with new ingredients which come from the observability inequality associated with the linear problem. When a stronger damping is in place, we show that the energy functional decays exponentially to zero and for this purpose a forced smoothing effect due to Aloui (Collect Math 59(1):53-62, 2008) plays an essential role in the proof. [ABSTRACT FROM AUTHOR]