  We establish various $L^{p}$ estimates for the Schr\"odinger operator
$-\Delta+V$ on Riemannian manifolds satisfying the doubling property and a
Poincar\'e inequality, where $\Delta $ is the Laplace-Beltrami operator and $V$
belongs to a reverse H\"{o}lder class. At the end of this paper we apply our
result on Lie groups with polynomial growth.
