Some differentiable sphere theorems

  • Qing Cui
  • Linlin SunEmail author


In this paper, we obtain several new intrinsic and extrinsic differentiable sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected \(n(\ge 4)\)-dimensional Riemannian manifold M is diffeomorphic to \(\mathbb {S}^n\) if one of the following conditions holds pointwisely:
$$\begin{aligned} (i)\ R_0>\left( 1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right) K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left( 1-\frac{6(\sqrt{10}-3)}{n-1}\right) K_{max}. \end{aligned}$$
Here \(K_{max}\), \(Ric^{[k]}\) and \(R_0\) stand for the maximal sectional curvature, the k-th weak Ricci curvature and the normalized scalar curvature. For extrinsic case, i.e., when M is a closed simply connected \(n(\ge 4)\)-dimensional submanifold immersed in \(\bar{M}\). We prove that M is diffeomorphic to \(\mathbb {S}^n\) if it satisfies some curvature pinching conditions. The only involved extrinsic quantities in our pinching conditions are the maximal sectional curvature \(\bar{K}_{max}\) and the squared norm of mean curvature vector \(\left|H\right|^2\). More precisely, we show that M is diffeomorphic to \(\mathbb {S}^n\) if one of the following conditions holds:
  1. (1)

    \(R_0\ge \left( 1-\frac{2}{n(n-1)}\right) \bar{K}_{max} +\frac{n(n-2)}{(n-1)^2}\left|H\right|^2\), and strict inequality is achieved at some point;

  2. (2)

    \(\dfrac{Ric^{[2]}}{2}\ge (n-2)\bar{K}_{max}+\frac{n^2}{8}\left|H\right|^2,\) and strict inequality is achieved at some point;

  3. (3)

    \(\dfrac{Ric^{[2]}}{2} \ge \frac{n(n-3)}{n-2}\left( \bar{K}_{max}+\left|H\right|^2\right) ,\) and strict inequality is achieved at some point.

It is worth pointing out that, in the proof of extrinsic case, we apply suitable complex orthonormal frame and simplify the calculations considerably. We also emphasize that both of the pinching constants in (2) and (3) are optimal for \(n=4\).

Mathematics Subject Classification

53C20 53C40 



We would like to thank Dr. Jun Sun for useful discussions and suggestions. We also want to thank the referee for his/her careful reading and useful comments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsSouthwest Jiaotong UniversityChengduChina
  2. 2.School of Mathematics and Statistics and Computational Science Hubei Key LaboratoryWuhan UniversityWuhanChina

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