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## A gap phenomenon on Riemannian manifolds with reverse volume pinching

Acta Mathematica Hungarica
Author:
P. Wang

## Abstract

Let M n be an n(≧ 3)-dimensional compact, simply connected Riemannian manifold without boundary and S n be the unit sphere of the Euclidean space R n+1. By two different means we derive an estimate of the diameter whenever the manifold considered satisfies that the sectional curvature K M ≦ 1, while Ric (M) ≧
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{{n + 2}}{4}$$ \end{document}
and the volume V (M) ≦
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{3}{2}$$ \end{document}
(1 + η)V (S n ) for some positive number η depending only on n. Consequently, a gap phenomenon of the manifold will be given according to the estimate of the diameter.
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## A differentiable sphere theorem on manifolds with reverse volume pinching

Acta Mathematica Hungarica
Author:
P. Wang

## Abstract

Based on the celebrated 1/4-pinching sphere theorem, we prove a differentiable sphere theorem on Riemannian manifolds with reverse volume pinching.

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