Harmonic forms on manifolds with non-negative Bakry-Aparts per thousandmery-Ricci curvature

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry-A parts per thousand mery-Ricci curvature if the space of weighted L (2) harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface.