In this paper, we consider a class of left invariant Riemannian metrics on Sp(n), which is invariant under the adjoint action of the subgroup Sp(n-3)×Sp(1)×Sp(1)×Sp(1). Based on the related formulae in the literature, we show that the existence of Einstein metrics is equivalent to the existence of solutions of some homogeneous Einstein equations. Then we use a technique of the Gröbner basis to get a sufficient condition for the existence, and show that this method will lead to new non-naturally reductive metrics.
Shaoxiang ZHANG
,
Huibin CHEN
,
Shaoqiang DENG
. NEW NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON Sp(n)[J]. Acta mathematica scientia, Series B, 2021
, 41(3)
: 887
-898
.
DOI: 10.1007/s10473-021-0315-x
[1] Arvanitoyeorgos A, Dzhepko V V, Nikonorov Y G. Invariant Einstein metrics on some homogeneous spaces of classical Lie groups. Canad J Math, 2009, 61(6):51-61
[2] Arvanitoyeorgos A, Mori K, Sakane Y. Einstein metrics on compact Lie groups which are not naturally reductive. Geom Dedicate, 2012, 160(1):261-285
[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
[4] Arvanitoyeorgos A, Sakane Y, Statha M. Einstein metrics on the symmetric group which are not naturally reductive//Current Developments in Differential Geometry and its Related Fields. Proceedings of the 4th International Colloquium on Differential Geometry and its Related Fields, Velico Tarnovo, Bulgaria 2014. World Scientific, 2015:1-22
[5] Besse A L. Einstein Manifolds. Berlin:Springer-Verlag, 1986
[6] Bohm C. Homogeneous Einstein metrics and simplicial complexes. J Differential Geom, 2004, 67(1):74-165
[7] Bohm C, Wang M, Ziller W. A variational approach for compact homogeneous Einstein manifolds. Geom Func Anal, 2004, 14(4):681-733
[8] Chen H B, Chen Z Q, Deng S Q. New non-naturally reductive Einstein metrics on Exceptional simple Lie groups. J Geom Phys, 2018, 124:268-285
[9] Chen Z Q, Chen H B. Non-naturally reductive Einstein metrics on Sp(n). Front Math China, 2020, 15(1):47-55
[10] Chen Z Q, Liang K. Non-naturally reductive Einstein metrics on the compact simple Lie group F4. Ann Glob Anal Geom, 2014, 46:103-115
[11] Chrysikos I, Sakane Y. Non-naturally reductive Einstein metrics on exceptional Lie groups. J Geom Phys, 2017, 116:152-186
[12] D' Atri J E, Ziller W. Naturally reductive metrics and Einstein metrics on compact Lie groups. Memoirs Amer Math Soc, 1979, 18(215):1-73
[13] Mori K. Left Invariant Einstein Metrics on SU(n) that are not naturally reductive[Master Thesis]. (in Japanese) Osaka University, 1994; English Translation:Osaka University RPM 96010(preprint series), 1996
[14] Park J S, Sakane Y. Invariant Einstein metrics on certain homogeneous spaces. Tokyo J Math, 1997, 20(1):51-61
[15] Wang M. Einstein metrics from symmetry and bundle constructions//Surveys in Differential Geometry:Essays on Einstein Manifolds. Surv Differ Geom VI. Boston, Ma:Int Press, 1999
[16] Wang M. Einstein metrics from symmetry and bundle constructions:A sequel//Differential Geometry:Under the Influence of S.-S. Chern. Advanced Lectures in Mathematics. Higher Education Press/International Press, 2012, 22:253-309
[17] Wang M, Ziller W. Existence and non-existence of homogemeous Einstein metrics. Invent Math, 1986, 84:177-194
[18] Yan Z L, Deng S Q. Einstein metrics on compact simple Lie groups attached to standard triples. Trans Amer Math Soc, 2017, 369:8587-8605
