Let (X, g) be a compact n-dimensional smooth Riemannian manifold with a lower bound on the average of the lowest \(n-p\) eigenvalues of the curvature operator and the diameter of X is bounded above by \(D>0\) . In this article, we investigate the relationship between the curvature operator and the Euler number of X. Our analysis is based on more general vanishing theorems for a Dirac operator associated with a smooth 1-form on X. As a consequence, we obtain partial affirmative answers to Question 4.6 posed by Herrmann et al. in (Ann Glob Anal Geom 44:391–399, 2013). Specifically, we prove that if a compact 2m-dimensional manifold admits an almost nonnegative curvature operator (ANCO) and has a nontrivial first de Rham cohomology group, then its Euler number vanishes. Furthermore, in the case where \(m=2\) , we show that the Euler number is nonnegative. This result provides a complete resolution to their question in the four-dimensional setting.
