Invariant Einstein metrics on generalized Wallach spaces

Abstract

Invariant Einstein metrics on generalized Wallach spaces have been classified except SO(k + l + m)/SO(k) × SO(l) × SO(m). In this paper, we give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type SO(k + l + m)/SO(k) × SO(l) × SO(m) admitting exactly two, three, or four invariant Einstein metrics up to a homothety.

Invariant Einstein metrics on Ledger–Obata spaces

“…Note also that F 4 / diag(F ) (with special choices of Riemannian metrics) could be considered as a generalized Wallach space (or a three locally symmetric space), see [3,14,15]. Hence, our results complete the classification of invariant Einstein metrics on generalized Wallach spaces [4].…”

On the stability of homogeneous Einstein manifolds II

“…The generalized Wallach spaces with $$G$$ simple are listed in Table 1, together with the numbers [123] and $$a_i:=[123]/d_i$$, where $$d_i:=\dim {\mathfrak {p} _i}$$ (see [14, 34]). For a table with the dimensions functions $$d_i$$, see [15, Table 1].…”

“…$$G$$‐stability is of course a necessary condition for the classical stability described above, and it is also extremely rare. As far as we know, besides irreducible symmetric metrics and the special case when any subalgebra of $$\mathfrak {g}$$ containing $$\mathfrak {k}$$ is of the form $$\mathfrak {k} \oplus \mathfrak {a}$$ with $$[\mathfrak {a} ,\mathfrak {a} ]=0$$ (for example, if $$K$$ is a maximal subgroup of $$G$$, see [9, 39]), the only known $$G$$‐stable Einstein metrics with $$\dim {\mathcal {M}_1^G}>1$$ are: the standard metric on $$\mathrm{SU}(2)$$, $$E_7/\mathrm{SO}(8)$$ and $$E_8/\mathrm{Spin}(8)\times \mathrm{Spin}(8)$$, discovered in [1] (see also [15, Remark 2.3]); and the unique Kähler–Einstein metric on each of the 13 flag manifolds with second Betti number $$b_2(M)$$ equal to 1, which can be deduced from [6, Theorem 3.1] (here …”

On locally homogeneous compact pseudo-Riemannian manifolds

“…so(2p, 1) su(p, 1) su(p) so(2, 2(n − 1)) so(1, 2(n − 1)) su(1, n − 1) su(n − p) so(4(n − 1), 4) so(4(n − 1), 3) sp(n − 1, 1) sp(p) so(4(n − 1), 4) so(4(n − 1), 3) sp(n − 1, 1) sp(n − 1) × sp(1) so (8,8) so ( Let us make the following observation. Assume that a reductive subgroup L acts properly and co-compactly on G/H.…”

“…This is a generalized Wallach space of the form SO(k + l + m)/SO(k) × SO(l) × SO(m) with m = 1. We refer to [8] for the existence of Einstein metrics.…”