About left-invariant geometry and homogeneous pseudo-Riemannian Einstein structures on the Lie group SU(3)

TL;DR

This paper explores left-invariant pseudo-Riemannian Einstein metrics on SU(3), discovering new Lorentzian examples and analyzing their geometric properties, including isometry groups and spectral characteristics relevant to physics.

Contribution

It presents the first known Lorentzian homogeneous Einstein metric on SU(3) and provides detailed analysis of invariant metrics, isometry groups, and spectral properties.

Findings

Recovered known Riemannian Einstein metrics on SU(3)

Discovered a new Lorentzian Einstein metric with positive constant

Analyzed the spectrum of the Laplacian for specific invariant metrics

Abstract

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.