Decomposition of the Curvature Operator and Applications to the Hopf Conjecture

TL;DR

This paper decomposes the curvature operator to establish eigenvalue criteria and vanishing theorems, confirming the Hopf conjecture in cases with small Weyl curvature and deriving new rigidity results.

Contribution

It introduces a novel decomposition of the curvature operator into Hermitian parts, leading to new criteria and theorems related to the Hopf conjecture and curvature conditions.

Findings

Confirmed the Hopf conjecture for manifolds with small Weyl curvature.

Established vanishing theorems for Betti numbers under integral bounds.

Derived rigidity theorems under harmonic Weyl curvature conditions.

Abstract

In this article, we investigate the interplay between the curvature operator, Weyl curvature, and the Hopf conjecture on compact Riemannian manifolds of even dimension. By decomposing the curvature operator into Hermitian components, we develop eigenvalue criteria for sectional curvature and prove vanishing theorems for Betti numbers under integral bounds on the Weyl tensor. Our results confirm the Hopf conjecture for manifolds with sufficiently small Weyl curvature, including locally conformally flat cases, and provide new rigidity theorems under harmonic Weyl curvature conditions.