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Riemannian and Randers Einstein Metrics on SO(n) Which Are Non-naturally Reductive

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Abstract

In this article, we prove that there are at least two new non-naturally reductive Ad(SO(l) × SO(k) × SO(k) × SO(k))-invariant Einstein metrics on compact simple Lie group SO(l + 3k) (k < l). It implies that every compact simple Lie group SO(n) (n > 12) admits at least \(2\left(\left[{n-1\over{4}}\right]-2\right)\) non-naturally reductive left invariant Einstein metrics. Moreover, we obtain that there are at least two families of invariant Einstein–Randers metrics on the compact Lie group SO(n) (n > 12). Besides, we construct non-naturally reductive left invariant (α, β) metrics on the Lie group SO(n) (n > 12). Finally, we examine the isometric problem for those Riemannian Einstein metrics.

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References

  1. Arvanitoyeorgos A., Dzhepko V.V., Nikonorov Yu.G., Invariant Einstein metrics on some homogeneous spaces of classical Lie groups. Canad. J. Math., 2009, 61(6): 1201–1213

    Article  MathSciNet  Google Scholar 

  2. Arvanitoyeorgos A., Mori K., Sakane Y., Einstein metrics on compact Lie groups which are not naturally reductive. Geom. Dedicata, 2012, 160(1): 261–285

    Article  MathSciNet  Google Scholar 

  3. Arvanitoyeorgos A., Sakane Y., Statha M., New Einstein metrics on the Lie group SO(n) which are not naturally reductive. Geom. Imaging Comput., 2015, 2(2): 77–108

    Article  MathSciNet  Google Scholar 

  4. Bao D., Chern S.S., Shen Z., An Introduction to Riemann–Finsler Geometry. New York: Springer-Verlag, 2000

    Book  Google Scholar 

  5. Bao D., Robles C., Ricci and flag curvatures in Finsler geometry. In: A Sampler of Riemann–Finsler Geometry, Math. Sci. Res. Inst. Publ., 50, Cambridge: Cambridge Univ. Press, 2004, 197–259

    Google Scholar 

  6. Besse A.L., Einstein Manifolds. Berlin: Springer-Verlag, 1987

    Book  Google Scholar 

  7. Chen H., Chen Z., Deng S., New non-naturally reductive Einstein metrics on exceptional simple Lie groups. J. Geom. Phys., 2018, 124: 268–285

    Article  MathSciNet  Google Scholar 

  8. Chen H., Chen Z., Deng S., Non-naturally reductive Einstein metrics on SO(n). Manuscripta Math., 2018, 156(1–2): 127–136

    Article  MathSciNet  Google Scholar 

  9. Chen Z., Chen H., Non-naturally reductive Einstein metrics on Sp(n). Front. Math. China, 2020, 15(1): 47–55

    Article  MathSciNet  Google Scholar 

  10. Chen Z., Liang K., Non-naturally reductive Einstein metrics on the compact simple Lie group F4. Ann. Global Anal. Geom., 2014, 46(2): 103–115

    Article  MathSciNet  Google Scholar 

  11. Chern S.S., Shen Z., Riemann–Finsler Geometry. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2005

    Book  Google Scholar 

  12. Chrysikos I., Sakane Y., Non-naturally reductive Einstein metrics on exceptional Lie groups. J. Geom. Phys., 2017, 116: 152–186

    Article  MathSciNet  Google Scholar 

  13. D’Atri J.E., Ziller W., Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Amer. Math. Soc., 1979, 18(215): iii+72 pp.

    MathSciNet  Google Scholar 

  14. Deng S., Hou Z., Homogeneous Einstein-Randers spaces of negative Ricci curvature. C. R. Math. Acad. Sci. Paris, 2009, 347(19–20): 1169–1172

    Article  MathSciNet  Google Scholar 

  15. Deng S., Hou Z., Naturally reductive homogeneous Finsler spaces. Manuscripta Math., 2010, 131(1–2): 215–229

    Article  MathSciNet  Google Scholar 

  16. Huang L., On the fundamental equations of homogeneous Finsler spaces. Differential Geom. Appl., 2015, 40: 187–208

    Article  MathSciNet  Google Scholar 

  17. Latifi D., Homogeneous geodesics in homogeneous Finsler spaces. J. Geom. Phys., 2007, 57(5): 1421–1433

    Article  MathSciNet  Google Scholar 

  18. Mori K., Left invariant Einstein metrics on SU(n) that are not naturally reductive. Master Thesis, Osaka: Osaka University, 1994 (in Japanese) (English Translation: Osaka University RPM 96010 (preprint series), 1996)

    Google Scholar 

  19. Parhizkar M., Naturally reductive homogeneous Finsler spaces. Vietnam J. Math., 2022, 50(1): 205–215

    Article  MathSciNet  Google Scholar 

  20. Park J.-S., Sakane Y., Invariant Einstein metrics on certain homogeneous spaces. Tokyo J. Math., 1997, 20(1): 51–61

    Article  MathSciNet  Google Scholar 

  21. Randers G., On an asymmetrical metric in the fourspace of general relativity. Phys. Rev. (2), 1941, 59: 195–199

    Article  MathSciNet  Google Scholar 

  22. Tan J., Xu M., Naturally reductive (α1,α2) metrics. Acta Math. Sci. Ser. B (Engl. Ed.), 2023, 43(4): 1547–1560

    MathSciNet  Google Scholar 

  23. Wang M., Einstein metrics from symmetry and bundle constructions. In: Surveys in Differential Geometry: Essays on Einstein Manifolds, Surv. Differ. Geom., 6, Boston, MA: Int. Press, 1999, 287–325

    Google Scholar 

  24. Yan Z., Deng S., Einstein metrics on compact simple Lie groups attached to standard triples. Trans. Amer. Math. Soc., 2017, 369(12): 8587–8605

    Article  MathSciNet  Google Scholar 

  25. Zhang B., Chen H., Tan J., New non-naturally reductive Einstein metrics on SO(n). Internat. J. Math., 2018, 29(11): 1850083, 13 pp.

    MathSciNet  Google Scholar 

  26. Zhang S., Chen H., New Einstein metrics on Sp(n) which are non-naturally reductive. Czechoslovak Math. J., 2022, 72(2): 349–363

    Article  MathSciNet  Google Scholar 

  27. Zhang S., Chen H., Deng S., New non-naturally reductive Einstein metrics on Sp(n). Acta Math. Sci. Ser. B (Engl. Ed.), 2021, 41(3): 887–898

    MathSciNet  Google Scholar 

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Acknowledgements

This paper is supported by National Natural Science Foundation of China (No. 12001007) and Natural Science Foundation of Anhui province (No. 1908085QA03).

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Authors and Affiliations

  1. School of Microelectronics and Data Science, Anhui University of Technology, Ma’anshan, 243032, China

    Bichao Sun & Ju Tan

  2. Anhui Provincial Joint Key Laboratory of Disciplines for Industrial Big Data Analysis and Intelligent Decision, Ma’anshan, 243032, China

    Ju Tan

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  1. Bichao Sun
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  2. Ju Tan
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Correspondence to Ju Tan.

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Sun, B., Tan, J. Riemannian and Randers Einstein Metrics on SO(n) Which Are Non-naturally Reductive. Front. Math (2024). https://doi.org/10.1007/s11464-023-0096-8

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  • DOI: https://doi.org/10.1007/s11464-023-0096-8

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