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FOUR-MANIFOLDS WITH POSITIVE CURVATURE

Part of: Global differential geometry

Published online by Cambridge University Press:  06 April 2020

R. DIÓGENES ,
E. RIBEIRO  and
E. RUFINO
R. DIÓGENES
Affiliation:
UNILAB, Instituto de Ciências Exatas e da Natureza, Campus dos Palmares, ROD. CE 060, KM 51, 62.785-000 Acarape, CE, Brazil e-mail: rafaeldiogenes@unilab.edu.br
E. RIBEIRO
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará - UFC, CAMPUS do Pici, Av. Humberto Monte, Bloco 914, 60455-760Fortaleza, CE, Brazil e-mail: ernani@mat.ufc.br
E. RUFINO
Affiliation:
Departamento de Matemática, Universidade Federal de Roraima - UFRR, Campus Paricarana, Av. CAP. Ene Garcez, 2413, 69310-000Boa Vista, RR, Brazil e-mail: elzimar.rufino@ufrr.br
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Abstract

In this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M4 is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$ , provided that the sectional curvatures all lie in the interval $\left[ {{{3\sqrt {3 - 5} } \over 4}, 1} \right]$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.

MSC classification

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C20: Global Riemannian geometry, including pinching 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Secondary: 53C65: Integral geometry
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 245 - 257
Copyright
© Glasgow Mathematical Journal Trust 2020

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Footnotes

E. Ribeiro Jr. was partially supported by grants from CNPq/Brazil (Grant: 303091/2015-0), PRONEX-FUNCAP/CNPq/Brazil, and CAPES/Brazil – Finance Code 001.

E. Rufino was partially supported by CAPES/Brazil.

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