Lp boundedness of Riesz transform related to Schroedinger operators on a manifold

We establish various L p estimates for the Schr¨odinger operator −
+V on Riemannian manifolds satisfying the doubling property and a Poincar´e inequality, where
is the Laplace-Beltrami operator and V belongs to a reverse H¨older class. At the end of this paper we apply our result to Lie groups with polynomial growth.