Manifolds with harmonic curvature and curvature operator of the second kind

TL;DR

This paper proves that complete Riemannian manifolds with harmonic curvature and certain nonnegative curvature operator conditions are necessarily Einstein or of constant curvature, extending previous results in geometric analysis.

Contribution

It establishes new rigidity results for manifolds with harmonic curvature under specific nonnegativity conditions on the curvature operator of the second kind.

Findings

Manifolds with harmonic curvature and specific curvature operator bounds are Einstein.

Complete Einstein manifolds with certain curvature bounds are of constant curvature.

Generalizes previous work by Dai-Fu on curvature conditions.

Abstract

We prove that complete Riemannian manifolds of dimension $n\ge3$ with harmonic curvature and $\frac{n(n+2)}{2(n+1)}$-nonnegative curvature operator of the second kind must be Einstein. In particular, We show that complete Einstein manifolds of dimension $n\ge4$ with $\frac{3n(n-1)^2(n+2)}{2(5n^3+3n^2-30n+16)}$-nonnegative curvature operator of the second kind must be of constant curvature, which generalizes the work of Dai-Fu \cite{DF}.