Invariant Einstein metrics on basic classical Lie supergroups

2025/08/28 by Huihui An, An, Huihui, Zaili Yan +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2508.20639

openalex publication_date 2025/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over ℝ. We consider a natural family of left invariant metrics parameterized by scaling factors on the simple and Abelian components of the reductive even part, using the canonical bi-invariant bilinear form. Explicit expressions for the Levi-Civita connection and Ricci tensor are derived, and the Einstein condition is reduced to a solvable algebraic system. Our main result shows that, except for the cases of A(m,n) with m≠ n, F(4), and their real forms, every real basic classical Lie superalgebra admits at least two distinct Einstein metrics. Notably, for D(n+1,n) and D(2,1;α), we obtain both Ricci flat and non Ricci flat Einstein metrics, a phenomenon not observed in the non-super setting.

Citations

• Homogeneous Einstein Metrics and Butterflies
• Non-compact Einstein manifolds with symmetry
• Einstein solvmanifolds are standard
• Naturally reductive metrics and Einstein metrics on compact Lie groups

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