Geometric structures on Lie groups with flat bi-invariant metric

2009/07/31 by Vicente Cortés, Cortés, Vicente, Lars Schäfer +1
Mathematics · Physics and Astronomy · #53C15 #53C50 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C15 #msc:53C50

paper · pdf · doi:10.48550/arxiv.0907.5492

to appear in Journal of Lie Theory

arxiv created 2009/07/31 · openalex publication_date 2009/07/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let L⊂ V=\bR^k,l be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k,l) is 2-step nilpotent and is defined by an element η∈ Λ³L⊂ Λ³V. If ηis of type (3,0)+(0,3) with respect to a skew-symmetric endomorphism J with J²=\e Id, then the Lie group \cal L(η) is endowed with a left-invariant nearly Kähler structure if \e =-1 and with a left-invariant nearly para-Kähler structure if \e =+1. This construction exhausts all complete simply connected flat nearly (para-)Kähler manifolds. If η≠ 0 has rational coefficients with respect to some basis, then \cal L(η) admits a lattice Γ, and the quotient Γ∖ \cal L(η) is a compact inhomogeneous nearly (para-)Kähler manifold. The first non-trivial example occurs in six dimensions.

Citations

• Curvatures of left invariant metrics on lie groups
• Homogeneous para-Kähler Einstein manifolds

Related

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• Nearly pseudo-Kähler and nearly para-Kähler six-manifolds
• Pseudoriemannian 2-Step Nilpotent Lie Groups
• A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups
• Geometric structures on nilpotent Lie groups: on their classification and a distinguished compatible metric