  {\bf Abstract} \,\, We consider the following nonlinear Schr\"{o}dinger
equation on exterior domain. \begin{equation} \begin{cases} iu_t+\Delta_g u +
ia(x)u - |u|^{p-1}u = 0 \qquad (x,t) \in \Omega\times (0,+\infty), \qquad
(1)\cr u\big|_\Gamma = 0\qquad t \in (0,+\infty), \cr u(x,0) = u_0(x)\qquad x
\in \Omega, \end{cases} \end{equation} where $1<p<\frac{n+2}{n-2}$,
$\Omega\subset\mathbb{R}^n$ ($n\ge3$) is an exterior domain and
$(\mathbb{R}^n,g)$ is a complete Riemannian manifold. We establish Morawetz
estimates for the system (1) without dissipation ($a(x)\equiv 0$ in (1)) and
meanwhile prove exponential stability of the system (1) with a dissipation
effective on a neighborhood of the infinity.
  It is worth mentioning that our results are different from the existing
studies. First, Morawetz estimates for the system (1) are directly derived from
the metric $g$ and are independent on the assumption of an (asymptotically)
Euclidean metric. In addition, we not only prove exponential stability of the
system (1) with non-uniform energy decay rate, which is dependent on the
initial data, but also prove exponential stability of the system (1) with
uniform energy decay rate. The main methods are the development of Morawetz
multipliers in non (asymptotically) Euclidean spaces and compactness-uniqueness
arguments.
