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

## TL;DR

This paper proves that on certain complete Riemannian manifolds with a parallel 1-form, all L^2-harmonic forms vanish, extending vanishing theorems in Morse-Novikov cohomology.

## Contribution

It establishes a vanishing theorem for L^2 Morse-Novikov cohomology on manifolds with a parallel 1-form, including Vaisman manifolds.

## Key findings

- L^2-harmonic forms are identically zero on the considered manifolds.
- The vanishing theorem applies to manifolds with a parallel 1-form, such as Vaisman manifolds.

## 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.