Homogeneous Einstein Metrics on SO(n)

2010/01/18 by C.N. Pope, Pope, C. N.
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #High Energy Physics - Theory (hep-th)

paper · pdf · doi:10.48550/arxiv.1001.2776

openalex publication_date 2010/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well known that every compact simple Lie group G admits an Einstein metric that is invariant under the independent left and right actions of G. In addition to this bi-invariant metric, with G x G symmetry, it was shown by D'Atri and Ziller that every compact simple Lie group except SU(2) and SO(3) admits at least one further homogeneous Einstein metric, invariant under G x H, where H is some proper subgroup of G. In this paper we consider the Lie groups G=SO(n) for arbitrary n, and provide an explicit construction of (3k-4) inequivalent homogeneous Einstein metrics on SO(2k), and (3k-3) inequivalent homogeneous Einstein metrics on SO(2k+1).

Citations

• Naturally reductive metrics and Einstein metrics on compact Lie groups

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