Scalar Curvature and Betti Numbers of Compact Riemannian Manifolds

Let M be a compact oriented Riemannian manifolds with positive scalar curvature. We first prove a vanishing theorem for p-th Betti number of M, by assuming that the norm of the concircular curvature is less than some positive multiple of the scalar curvature at each point. In the second part, we show that if M has positive scalar curvature, then the existence of non-trivial harmonic p-forms imposes a certain integral inequality concerning the scalar curvature and the traceless Ricci curvature. Moreover, we provide a metric characterization of the equality case.