# Morse-Novikov cohomology of almost nonnegatively curved manifolds

## TL;DR

This paper proves that for certain almost nonnegatively curved manifolds with nonzero first cohomology, the Morse-Novikov cohomology groups vanish, extending to Ricci curvature under bounded curvature operator conditions.

## Contribution

It establishes vanishing results for Morse-Novikov cohomology on almost nonnegatively curved manifolds with nonzero first de Rham cohomology, including Ricci curvature cases.

## Key findings

- Morse-Novikov cohomology vanishes for manifolds with almost nonnegative sectional curvature and nonzero first cohomology.
- Vanishing also holds for manifolds with almost nonnegative Ricci curvature when curvature operator is bounded below.
- Results extend classical cohomology vanishing theorems to the Morse-Novikov setting under curvature conditions.

## Abstract

Let $M^n$ be a closed manifold of almost nonnegative sectional curvature and nonzero first de Rham cohomology group. For any $[\theta] \in H^1_{dR}(M^n), [\theta] \neq 0$, we show that the Morse- Novikov cohomology group $H^p(M^n, \theta)$ vanishes for any $p$. A similar result holds for a closed manifold of almost nonnegative Ricci curvature under the additional assumption that its curvature operator is uniformly bounded from below.