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

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Abstract

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

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    PeiHe Wang

  2. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    YuLiang Wen

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  1. PeiHe Wang
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  2. YuLiang Wen
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Correspondence to PeiHe Wang.

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Wang, P., Wen, Y. A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching. Sci. China Math. 54, 603–610 (2011). https://doi.org/10.1007/s11425-010-4126-0

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  • DOI: https://doi.org/10.1007/s11425-010-4126-0

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