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Non-naturally reductive Einstein metrics on Sp(n)

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Abstract

We prove that Sp(2k + l) admits at least two non-naturally reductive Einstein metrics which are Ad(Sp(k) × Sp(k) × Sp(l))-invariant if k < l. It implies that every compact simple Lie group Sp(n) for n ⩾ 4 admits at least 2[(n–1)/3] non-naturally reductive left-invariant Einstein metrics.

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References

  1. 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 

  2. Arvanitoyeorgos A, Sakane Y, Statha M. Einstein metrics on the symplectic group which are not naturally reductive. In: Adachi T, Hashimoto H, Hristov M J, eds. Current Developments in Differential Geometry and its Related Fields. Singapore: World Scientific, 2016, 1–21

    MATH  Google Scholar 

  3. Besse A L. Einstein Manifolds. Berlin: Springer-Verlag, 1986

    MATH  Google Scholar 

  4. 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 

  5. Chen Z, Kang Y, Liang K. Invariant Einstein metrics on three-locally-symmetric spaces. Comm Anal Geom, 2016, 24(4): 769–792

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. D’Atri J E, Ziller W. Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups. Mem Amer Math Soc, Vol 18, No 215. Providence: Amer Math Soc, 1979

    Google Scholar 

  9. Jensen G. Einstein metrics on principle fiber bundles. J Differential Geom, 1973, 8: 599–614

    Article  Google Scholar 

  10. Mori K. Left Invariant Einstein Metrics on SU(n) that are not Naturally Reductive. Master Thesis, Osaka University, 1994 (in Japanese); Osaka University RPM 96010 (Preprint Ser), 1996

    Google Scholar 

  11. Nikonorov Yu G. On a class of homogeneous compact Einstein manifolds. Sib Math J, 2000, 41: 168–172

    Article  MathSciNet  Google Scholar 

  12. Nikonorov Yu G. Classification of generalized Wallach spaces. Geom Dedicata, 2016, 181: 193–212

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Wang M, Ziller W. Existence and non-existence of homogeneous Einstein metrics. Invent Math, 1986, 84: 177–194

    Article  MathSciNet  Google Scholar 

  15. 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 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Nos. 11571182, 11901300, 11931009), the Natural Science Foundation of Tian** (No. 19JCYBJC30600), and the Natural Science Research of Jiangsu Education Institutions of China (No. 19KJB110015).

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Authors and Affiliations

  1. School of Mathematical Sciences, Nankai University, Tian**, 300071, China

    Zhiqi Chen

  2. School of Mathematical Sciences, Nan**g Normal University, Nan**g, 210023, China

    Huibin Chen

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  1. Zhiqi Chen
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  2. Huibin Chen
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Correspondence to Huibin Chen.

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Chen, Z., Chen, H. Non-naturally reductive Einstein metrics on Sp(n). Front. Math. China 15, 47–55 (2020). https://doi.org/10.1007/s11464-020-0818-0

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  • DOI: https://doi.org/10.1007/s11464-020-0818-0

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