Morse-Novikov cohomology of almost nonnegatively curved manifolds

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.