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Four-dimensional manifolds with pinched positive sectional curvature

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Abstract

We prove that a four-dimensional compact oriented connected Riemannian manifold whose sectional curvatures all lie in the interval \([\frac{1}{1+3\sqrt{3}}, 1]\) is necessarily definite. In particular, we improve the pinching constants considered by some preceding works. In addition, we show that a four-dimensional compact oriented Einstein manifold whose sectional curvatures all lie in the interval \([\frac{1}{10}, 1]\) is either topologically \(\mathbb {S}^4\) or homothetically isometric to \(\mathbb {CP}^2,\) equipped with its standard Fubini-Study metric.

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Acknowledgements

The authors want to thank the referee for his careful reading, relevant remarks and valuable suggestions. Moreover, the authors want to thank E. Costa for helpful conversations about this subject.

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Authors and Affiliations

  1. Instituto de Ciências Exatas e da Natureza, UNILAB, Acarape, CE, 62785-000, Brazil

    R. Diógenes

  2. Departamento de Matemática, Universidade Federal do Ceará - UFC, Campus do Pici, Av. Humberto Monte, Bloco 914, Fortaleza, CE, 60455-760, Brazil

    E. Ribeiro Jr.

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  1. R. Diógenes
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  2. E. Ribeiro Jr.
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Correspondence to E. Ribeiro Jr..

Additional information

R. Diógenes was partially supported by FUNCAP/Brazil. E. Ribeiro Jr. was partially supported by CNPq/Brazil, Grant: 303091/2015-0.

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Diógenes, R., Ribeiro, E. Four-dimensional manifolds with pinched positive sectional curvature. Geom Dedicata 200, 321–330 (2019). https://doi.org/10.1007/s10711-018-0373-y

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