Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption

Abstract

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\ge -Kg$ and $diam\le D$ for some positive constants $K>0$ and $D>0$.

77 pages v2: minor change v3: an application to the almost umbilical manifolds was added v3: some simplifications and improvements