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A NOTE ON THE CLASSIFICATION OF NONCOMPACT QUASI-EINSTEIN MANIFOLDS WITH VANISHING CONDITION ON THE WEYL TENSOR

Part of: Global differential geometry

Published online by Cambridge University Press:  07 May 2021

H. BALTAZAR  and
M. MATOS NETO
H. BALTAZAR
Affiliation:
Departamento de Matemática, Universidade Federal do Piauí 64049-550 Teresina, Piauí, Brazil e-mail: halyson@ufpi.edu.br,mvieira@ufpi.edu.br
M. MATOS NETO
Affiliation:
Departamento de Matemática, Universidade Federal do Piauí 64049-550 Teresina, Piauí, Brazil e-mail: halyson@ufpi.edu.br,mvieira@ufpi.edu.br
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Abstract

The aim of this paper is to study complete (noncompact) m-quasi-Einstein manifolds with λ=0 satisfying a fourth-order vanishing condition on the Weyl tensor and zero radial Weyl curvature. In this case, we are able to prove that an m-quasi-Einstein manifold (m>1) with λ=0 on a simply connected n-dimensional manifold(Mn, g), (n ≥ 4), of nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with (n–1)–dimensional Einstein fiber, provided that M has fourth-order divergence-free Weyl tensor (i.e. div4W =0).

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 64 , Issue 2 , May 2022 , pp. 336 - 346
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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