Non-naturally reductive Einstein metrics on Sp(n)

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.

Key words： Einstein metric; non-naturally reductive metric; Riemannian manifold; Lie group

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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