Einstein metrics on group manifolds and cosets

It is well known that every compact simple group manifold GG admits a bi-invariant Einstein metric, invariant under GL×GRGL×GR. Less well known is that every compact simple group manifold except SO(3)SO(3) and SU(2)SU(2) admits at least one more homogeneous Einstein metric, invariant still under GLGL but with some, or all, of the right-acting symmetry broken. (SO(3)SO(3) and SU(2)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)G=SU(3), SO(5)SO(5) and G2G2. For G=SU(3)G=SU(3), we find just the two already known inequivalent Einstein metrics. For G=SO(5)G=SO(5), we find four inequivalent Einstein metrics, thus extending previous results where only two were known. For G=G2G=G2 we find six inequivalent Einstein metrics, which extends the list beyond the previously-known two examples. We also study some cosets G/HG/H for the above groups GG. In particular, for SO(5)/U(1)SO(5)/U(1) we find, depending on the embedding of the U(1)U(1), generically two, with exceptionally one or three, Einstein metrics. We also find a pseudo-Riemannian Einstein metric of signature (2,6)(2,6) on SU(3)SU(3), an Einstein metric of signature (5,6)(5,6) on G2/SU(2)diag, and an Einstein metric of signature (4,6)(4,6) on G2/U(2)G2/U(2). Interestingly, there are no Lorentzian Einstein metrics among our examples.