$L^{p}$ Boundedness of Riesz transform related to Schr\"odinger operators on a manifold

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.