Let Mn be a compact, simply connected n (⩾ 3)-dimensional Riemannian manifold without boundary and Sn be the unit sphere Euclidean space ℝn+1. We derive a differentiable sphere theorem whenever the manifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M) ⩾ \( \frac{{n + 2}} {4} \) and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.

A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching

The authors introduce the Hausdorff convergence to discuss the differentiable sphere theorem with excess pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds is derived.

https://www.camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=30B106&flag=1&journal_id=camb&year_id=2010

A rigidity phenomenon on riemannian manifolds with reverse excess pinching

A differentiable sphere theorem with pinching integral ricci curvature

In the article stated here, we introduce the Hausdorff convergence to discuss the differentiable sphere theorem with some curvature conditions and volume pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds will be derived.

A rigidity phenomenon on Riemannian manifolds with reverse volume pinching

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A rigidity phenomenon on Riemannian manifolds with reverse volume pinching