We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless weighted $L^p$ estimate with optimal exponent. Other than previous arguments, only a small part of our proof is based on special auxiliary functions, the core of the argument is a weak type estimate and a sparse decomposition of the stochastic process by X.D. Li, whose projection is the Riesz vector.