m-Quasi-Einstein Metrics Satisfying Certain Conditions on the Potential Vector Field

Abstract

In this paper we study Riemannian manifolds \((M^n, g)\) admitting an m-quasi-Einstein metric with V as its potential vector field. We derive an integral formula for compact m-quasi-Einstein manifolds and prove that the vector field V vanishes under certain integral inequality. Next, we prove that if the metrically equivalent 1-form \(V^{\flat }\) associated with the potential vector field is a harmonic 1-form, then V is an infinitesimal harmonic transformation. Moreover, if M is compact then it is Einstein. Some more results were obtained when (i) V generates an infinitesimal harmonic transformation, (ii) V is a conformal vector field.

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Acknowledgements

The author is grateful to the reviewers for their valuable comments and suggestions.

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Correspondence to Amalendu Ghosh.

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Ghosh, A. m-Quasi-Einstein Metrics Satisfying Certain Conditions on the Potential Vector Field. Mediterr. J. Math. 17, 115 (2020). https://doi.org/10.1007/s00009-020-01558-8

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Keywords

  • Ricci soliton
  • m-quasi-Einstein metric
  • infinitesimal harmonic transformation
  • conformal vector field

Mathematics Subject Classification

  • 53C25
  • 53C20
  • 53D10