Vanishing theorems on complete Riemannian manifold with a parallel $1$-form

Abstract

In this article, we first consider the $L^2$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^2$ \textit{Morse-Novikov cohomology}, we prove that the $L^2$-harmonic forms on $M$ are identically zero.

13 pages; to appear in Mathematische Nachrichten