Rigidity theorem for compact Bach-flat manifolds with positive constant $\sigma_2$

TL;DR

This paper proves that compact Bach-flat manifolds with positive constant _2 and certain curvature conditions are necessarily Einstein, advancing understanding of geometric structures under curvature constraints.

Contribution

It establishes a rigidity theorem showing that under specific curvature pinching, Bach-flat manifolds with positive _2 are Einstein, a new result in geometric analysis.

Findings

Bach-flat manifolds with positive _2 are Einstein under curvature pinching.

Provides conditions under which Bach-flat manifolds exhibit Einstein geometry.

Advances classification of manifolds with special curvature properties.

Abstract

We prove that an n($\geq$ 4)-dimensional compact Bach-flat manifold with positive constant $\sigma_2$ is an Einstein manifold, provided that its Weyl curvature satisfies a suitable pinching condition.