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Rigid Properties of Generalized \(\tau \)-Quasi Ricci-Harmonic Metrics

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Abstract

In this paper, we study compact generalized \(\tau \)-quasi Ricci-harmonic metrics. In the first part, we explore conditions under which generalized \(\tau \)-quasi Ricci-harmonic metrics are harmonic-Einstein and give some characterization results for this case. In the second part, we obtain some rigidity results for compact \((\tau , \rho )\)-quasi Ricci-harmonic metrics which are a special case of generalized \(\tau \)-quasi Ricci-harmonic metrics. In the third part, we give two gap theorems for compact \(\tau \)-quasi Ricci-harmonic metrics by showing some necessary and sufficient conditions for the metrics to be harmonic-Einstein.

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References

  1. Besse, L.: Einstein Manifolds. Ergebnisse derMathematik und ihrerGrenzgebiete (3), vol. 10. SpringerBerlin (1987)

  2. Barros, A., Ribeiro Jr., E.: Characterizations and integral formulae for generalized \(m\)-quasi-Einstein metrics. Bull. Braz. Math. Soc. (N.S.) 45, 325–341 (2014)

    Article  MathSciNet  Google Scholar 

  3. Cao, H.: Recent progress on Ricci soliton. ALM 11, 1–38 (2009)

    MathSciNet  Google Scholar 

  4. Case, J., Shu, J., Wei, G.: Rigidity of quasi-Einstein metrics. Differ. Geom. Appl. 29, 93–100 (2011)

    Article  MathSciNet  Google Scholar 

  5. Catino, G.: Generalized quasi-Einstein manifolds with harmonic Weyl tensor. Math. Z. 271, 751–756 (2012)

    Article  MathSciNet  Google Scholar 

  6. Deng, Y.: A note on generalized quasi-Einstein manifolds. Math. Nachr. 288, 1122–1126 (2015)

    Article  MathSciNet  Google Scholar 

  7. Eells, J., Sampson, J.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–169 (1964)

    Article  MathSciNet  Google Scholar 

  8. Fernández-López, M., García-Río, E.: Some gap theorems for gradient Ricci solitons. Int. J. Math. 23, 9 (2012)

    Article  MathSciNet  Google Scholar 

  9. Guo, H., Philipowski, R., Thalmaier, A.: On gradient solitons of the Ricci-harmonic flow. Acta Math. Sin. (Engl. Ser.) 31, 1798–1804 (2015)

    Article  MathSciNet  Google Scholar 

  10. He, C., Petersen, P., Wylie, W.: On the classification of warped product Einstein metrics. Commun. Anal. Geom. 20, 271–312 (2012)

    Article  MathSciNet  Google Scholar 

  11. Hu, Z., Li, D., Xu, J.: On generalized \(m\)-quasi-Einstein manifolds with constant scalar curvature. J. Math. Anal. Appl. 432, 733–743 (2015)

    Article  MathSciNet  Google Scholar 

  12. Hu, Z., Li, D., Zhai, S.: On generalized \(m\)-quasi-Einstein manifolds with constant Ricci curvatures. J. Math. Anal. Appl. 446, 843–851 (2017)

    Article  MathSciNet  Google Scholar 

  13. Huang, G., Wei, Y.: The classification of \((m, \rho )\)-quasi-Einstein manifolds. Ann. Glob. Anal. Geom. 44, 269–282 (2013)

    Article  MathSciNet  Google Scholar 

  14. Huang, G., Li, H.: On a classification of the quasi-Yamabe gradient solitons. Methods Appl. Anal. 21, 379–389 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Huang, G., Zeng, F.: A note on gradient generalized quasi-Einstein manifolds. J. Geom. 106, 297–311 (2015)

    Article  MathSciNet  Google Scholar 

  16. Huang, G.: Integral pinched gradient shrinking \(\rho \)-Einstein solitons. J. Math. Anal. Appl. 451, 1045–1055 (2017)

    Article  MathSciNet  Google Scholar 

  17. Ma, B., Huang, G.: Lower bounds for the scalar curvature of noncompact gradient solitons of List’s flow. Arch. Math. (Basel) 100, 593–599 (2013)

    Article  MathSciNet  Google Scholar 

  18. Müller, R.: The Ricci Flow Coupled with Harmonic Map Heat Flow. Ph.D. thesis, ETH Zürich (2009). http://e-collection.library.ethz.ch//view/eth:41938

  19. Müller, R.: Ricci flow coupled with harmonic map flow. Ann. Sci. Éc. Norm. Supér. 45, 101–142 (2012)

    Article  MathSciNet  Google Scholar 

  20. Shin, J.: On the classification of \(4\)-dimensional \((m, \rho )\)-quasi-Einstein manifolds with harmonic Weyl curvature. Ann. Glob. Anal. Geom. 51, 379–399 (2017)

    Article  MathSciNet  Google Scholar 

  21. Tadano, H.: Gap theorems for Ricci-harmonic solitons. Ann. Glob. Anal. Geom. 49, 165–175 (2016)

    Article  MathSciNet  Google Scholar 

  22. Williams, M.: Results on coupled Ricci and harmonic map flows. Adv. Geom. 15, 7–26 (2015)

    Article  MathSciNet  Google Scholar 

  23. Wu, G., Zhang, S.: Volume growth of shrinking gradient Ricci-harmonic soliton. Results Math. 72, 205–223 (2017)

    Article  MathSciNet  Google Scholar 

  24. Wang, L.: Rigid properties of quasi-almost-Einstein metrics. Chin. Ann. Math. Ser. B 33, 715–736 (2012)

    Article  MathSciNet  Google Scholar 

  25. Wang, L.: On quasi Ricci-harmonic metrics. Ann. Acad. Sci. Fennic Math. 41, 417–437 (2016)

    Article  Google Scholar 

  26. Wang, L.: On \(f\)-non-parabolic ends for Ricci-harmonic metrics. Ann. Glob. Anal. Geom. 51, 91–107 (2017)

    Article  MathSciNet  Google Scholar 

  27. Wang, L.: On gradient quasi-Einstein solitons. J. Geom. Phys. 123, 484–494 (2018)

    Article  MathSciNet  Google Scholar 

  28. Wang, L.: Gap results for compact quasi-Einstein metrics. Sci. China Math. 61, 943–954 (2018)

    Article  MathSciNet  Google Scholar 

  29. Yang, F., Shen, J.: Volume growth for gradient shrinking solitons of Ricci-harmonic flow. Sci. China Math. 55, 1221–1228 (2012)

    Article  MathSciNet  Google Scholar 

  30. Zeng, F.: Rigidity of \(\tau \)-quasi Ricci-harmonic metrics. Indian J. Pure Appl. Math. 49, 431–449 (2018)

    Article  MathSciNet  Google Scholar 

  31. Zhu, M.: On the relation between Ricci-harmonic solitons and Ricci solitons. J. Math. Anal. Appl. 447, 882–889 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the referee for the careful reading of our manuscript. This work was supported by NSFC (Nos. 11971415, 11926501, 11901500), and Project of Henan Provincial Department of Education (21A110021) and Nanhu Scholars Program for Young Scholars of XYNU (Grant No. 2019).

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  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, People’s Republic of China

    Fanqi Zeng

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  1. Fanqi Zeng
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Zeng, F. Rigid Properties of Generalized \(\tau \)-Quasi Ricci-Harmonic Metrics. Results Math 75, 165 (2020). https://doi.org/10.1007/s00025-020-01299-w

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