Einstein Metrics on Group Manifolds and Cosets

TL;DR

This paper explores and classifies Einstein metrics on certain compact simple Lie groups and their cosets, discovering new inequivalent metrics and analyzing their properties, including signatures and symmetry invariance.

Contribution

It extends the known list of Einstein metrics on G=SO(5) and G=G_2, and provides new examples on cosets, including pseudo-Riemannian metrics with various signatures.

Findings

Found four inequivalent Einstein metrics on SO(5).

Discovered six inequivalent Einstein metrics on G_2.

Identified pseudo-Riemannian Einstein metrics with specific signatures.

Abstract

It is well known that every compact simple group manifold G admits a bi-invariant Einstein metric, invariant under G_L\times G_R. Less well known is that every compact simple group manifold except SO(3) and SU(2) admits at least one more homogeneous Einstein metric, invariant still under G_L but with some, or all, of the right-acting symmetry broken. (SO(3) and SU(2) are exceptional in admitting only the one, bi-invariant, Einstein metric.) In this paper, we look for Einstein metrics on three relatively low dimensional examples, namely G=SU(3), SO(5) and G_2. For G=SU(3), we find just the two already known inequivalent Einstein metrics. For G=SO(5), we find four inequivalent Einstein metrics, thus extending previous results where only two were known. For G=G_2 we find six inequivalent Einstein metrics, which extends the list beyond the previously-known two examples. We also study some cosets G/H for the above groups G. In particular, for SO(5)/U(1) we find, depending on the embedding of the U(1), generically two, with exceptionally one or three, Einstein metrics. We also find a pseudo-Riemannian Einstein metric of signature (2,6) on SU(3), an Einstein metric of signature (5,6) on G_2/SU(2)_{diag}, and an Einstein metric of signature (4,6) on G_2/U(2). Interestingly, there are no Lorentzian Einstein metrics among our examples.