Curvature operator and Euler number

TL;DR

This paper explores how curvature bounds influence the Euler number of compact Riemannian manifolds, establishing conditions under which the Euler number must vanish or be nonnegative, especially in four dimensions.

Contribution

It provides new vanishing theorems linking curvature operator bounds to the Euler number, answering a specific open question in four-dimensional geometry.

Findings

Euler number vanishes for manifolds with almost nonnegative curvature operator and nontrivial first cohomology.

In four dimensions, the Euler number is proven to be nonnegative under the given conditions.

Partial affirmative answers to a question by Herrmann, Sebastian, and Tuschmann are obtained.

Abstract

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, Sebastian, and Tuschmann in \cite{HST}. 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.