2021/07/26 by Robert Coquereaux, Coquereaux, Robert
Mathematics · Physics and Astronomy · #22E46 (Secondary) #53C30 #81R05 (Primary) 53C25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Group Theory (math.GR) #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2107.12285
openalex publication_date 2021/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a collection of notes on the properties of left-invariant metrics on the eight-dimensional compact Lie group SU(3). Among other topics we investigate the existence of invariant pseudo-Riemannian Einstein metrics on this manifold. We recover the known examples (Killing metric and Jensen metric) in the Riemannian case (signature (8,0)), as well as a Gibbons et al example of signature (6,2), and we describe a new example, which is Lorentzian (ie of signature (7,1). In the latter case the associated metric is left-invariant, with isometry group SU(3) x U(1), and has positive Einstein constant. It seems to be the first example of a Lorentzian homogeneous Einstein metric on this compact manifold. These notes are arranged into a paper that deals with various other subjects unrelated with the quest for Einstein metrics but that may be of independent interest: Among other topics we describe the various groups that may arise as isometry groups of left-invariant metrics on SU(3), provide parametrizations for these metrics, give several explicit results about the curvatures of the corresponding Levi-Civita connections, discuss modified Casimir operators (quadratic, but also cubic) and Laplace-Beltrami operators. In particular we discuss the spectrum of the Laplacian for metrics that are invariant under SU(3) x U(2), a subject that may be of interest in particle physics.