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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: [email protected],[email protected]
M. MATOS NETO
Affiliation:
Departamento de Matemática, Universidade Federal do Piauí 64049-550 Teresina, Piauí, Brazil e-mail: [email protected],[email protected]
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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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References

Bach, R., Zur Weylschen Relativiätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs, Math. Z. 9 (1921), 110–135.CrossRefGoogle Scholar
Bakry, D. and Ledoux, M., Sobolev inequalities and Myers diameter theorem for an abstract Markov generator, Duke Math. J. 85 (1996), 253–270.CrossRefGoogle Scholar
Barros, A., Batista, R. and Ribeiro, E. Jr., Bounds on volume growth of geodesic balls for Einstein warped products, Proc. Amer. Math. Soc. 143 (2015) 4415–4422.CrossRefGoogle Scholar
Barros, A., Ribeiro, E. Jr. and Silva, J., Uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous manifolds, Differ. Geom. Appl. 35 (2014), 60–73.CrossRefGoogle Scholar
Besse, A., Einstein manifolds (Springer-Verlag, Berlin, Heidelberg, 1987).CrossRefGoogle Scholar
Cao, H.-D., Recent progress on Ricci solitons, in Recent advances in geometric analysis (Taipei, 2007), Advanced Lectures in Mathematics (ALM), vol. 11 (International Press, Somerville, MA, 2010), 1–38.Google Scholar
Cao, H.-D., Geometry of complete gradient shrinking Ricci solitons, in Geometry and analysis, No. 1 (Cambridge, MA, 2008), Advanced Lectures in Mathematics (ALM), vol. 17, (International Press, Somerville, MA, 2011), 227–246.Google Scholar
Cao, H.-D. and Chen, Q., On locally conformally flat gradient steady Ricci solitons, Trans. Am. Math. Soc. 364 (2012), 2377–2391.CrossRefGoogle Scholar
Cao, H.-D. and Chen, Q., On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), 1149–1169.CrossRefGoogle Scholar
Cao, H.-D., Giovanni, C., Chen, Q., Mantegazza, C. and Mazzieri, L., On Bach-flat gradient steady Ricci solitons, Calc. Var. 49 (2014) 125–138.CrossRefGoogle Scholar
Case, J., Shu, Y. and Wei, G., Rigidity of quasi-Einstein metrics, Differ. Geom. Appl. 29 (2011), 93–100.CrossRefGoogle Scholar
Catino, G., A note on four dimensional (anti-)self-dual quasi-Einstein manifolds, Differ. Geom. Appl. 30 (2012), 660–664.CrossRefGoogle Scholar
Catino, G., Complete gradient shrinking Ricci solitons with pinched curvature, Math. Ann. 355 (2013), 629–635.CrossRefGoogle Scholar
Catino, G., Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012), 751–756.CrossRefGoogle Scholar
Catino, G., Matrolia, P. and Monticelli, D., Grandient Ricci solitons with vanishing condictions on Weyl, J. Math. Pures Appl. 108 (2017), 1–13.CrossRefGoogle Scholar
Catino, G., Mantegazza, C., Mazzieri, L. and Rimoldi, M., Locally conformally flat quasi-Einstein manifolds, J. Reine Angew. Math. 2013(675) (2013), 181–189.CrossRefGoogle Scholar
Chen, Q. and He, C., On Bach flat warped product Einstein manifolds, Pac. J. Math. 265 (2013), 313–326.CrossRefGoogle Scholar
Derdzinski, A., Self-dual Kähler manifold and Einstein manifold of dimension four, Compos. Math. 49 (1983), 405–433.Google Scholar
Fernández-López, M. and García-Río, E., Rigidity of shrinking Ricci solitons, Math. Z. 269 (2011), 461–466.CrossRefGoogle Scholar
He, C., Petersen, P. and Wylie, W., On the classification of warped product Einstein metrics, Commun. Anal. Geom. 20 (2012), 271–311.CrossRefGoogle Scholar
Kim, D. and Kim, Y., Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Am. Math. Soc. 131 (2003) 2573–2576.CrossRefGoogle Scholar
Leandro, B., Generalized quasi-Einstein manifolds with harmonic anti-self dual Weyl tensor, Arch. Math. 106 (2016), 489–499.Google Scholar
Munteanu, O. and Sesum, N., On gradient Ricci solitons, J. Geom. Anal. 23(2) (2013), 539–561.CrossRefGoogle Scholar
Petersen, P. and Wylie, W., On the classification of gradient ricci solitons, Geom. Topol. 14 (2010), 2277–2300.CrossRefGoogle Scholar
Qian, Z., Estimates for weighted volumes and applications, Quart. J. Math. 48 (1997), 235–242.CrossRefGoogle Scholar
Ranieri, M. and Ribeiro, E. Jr., Bach-flat noncompact steady quasi-Einstein manifold, Arch. Math. 108 (2017), 507–519.CrossRefGoogle Scholar
Rimoldi, M., A remark on Einstein warped products, Pac. J. Math. 252 (2011), 207–218.CrossRefGoogle Scholar
Rimoldi, M., Rigidity results for Lichnerowicz Bakry-Emery Ricci tensors, PhD Thesis (Universita degli Studi di Milano, 2011).Google Scholar
Wang, L., On noncompact τ–quasi-Einstein metrics, Pac. J. Math. 245 (2011), 449–464.CrossRefGoogle Scholar
Wei, G. and Wylie, W., Comparison geometry for the smooth metric measure spaces, in Proceedings of the 4th ICCM, vol. 2 (Higher Education Press, Beijing, 2007), 191–202.Google Scholar
Wei, G. and Wylie, W., Comparison geometry for the Bakry-Emery Ricci tensor, J. Differ. Geom. 83(2) (2009), 377–405.CrossRefGoogle Scholar
Yang, F., Wang, Z. and Zhang, L., On the classification on four-dimensional gradient Ricci solitons. arXiv: 1707.04846v1 [math.DG].Google Scholar
Yang, F. and Zhang, L., Rigidity of gradient shrinking Ricci solitons. arXiv: 1705.09754v1 [math.DG].Google Scholar
Yau, S., Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.CrossRefGoogle Scholar