Rigidity of complete Riemannian manifolds with vanishing Bach tensor

TL;DR

This paper establishes rigidity theorems for complete Riemannian manifolds with zero Bach tensor, positive scalar curvature, and specific curvature inequalities, revealing conditions under which the manifold's geometry is uniquely determined.

Contribution

It introduces new rigidity results for manifolds with vanishing Bach tensor, extending previous understanding through pointwise and integral curvature inequalities.

Findings

Rigidity characterized by pointwise inequalities for manifolds with vanishing Bach tensor.

Rigidity results under inequalities involving $L^{n/2}$-norms of curvature tensors.

Conditions under which the manifold's geometry is uniquely determined by curvature bounds.

Abstract

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.