On vanishing theorems for harmonic $p$-forms and Betti numbers of compact Riemannian manifolds

Abstract

Under the bounded condition for the integral of the traceless Ricci tensor, we obtain a vanishing theorem for harmonic $p$-forms and Betti numbers of compact oriented locally conformally flat Riemannian manifolds without satisfying a sign condition of the scalar curvature. In light of the preceding result, we show the relationship between the existence of non-trivial harmonic $p$-forms and a certain integral inequality of the scalar curvature and the traceless Ricci curvature. As a result, the structure of the given Riemannian manifold can be defined. Our results extend and improve the previous results in these directions.