Invariant Einstein Metrics on Generalized Flag Manifolds With Two Isotropy Summands

Abstract

Let M = G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.2010 Mathematics subject classification: primary 53C25; secondary 53C30, 22E46.

Invariant Einstein metrics on flag manifolds with four isotropy summands

“…Then, it is evident that m = m 2 ⊕ m 4 , and m is a linear subspace of m = ⊕ 4 k=1 m k . In [8], it was shown that (c 4 22 4 , so by using Table 5, we obtain the following values: …”

“…In our case, the fibers U /K are the spaces SO (9) (10), and E 7 /SU (7) × U (1), respectively. These are generalized flag manifolds with two isotropy summands (see [8]), so we can easily compute the triples (c 4 22 ) for the spaces U /K . Let B G = B and B U denote the Killing forms of G and U , respectively, and let u = k ⊕m be a reductive decomposition of u with respect to B U , where m ∼ = T eK (U /K ).…”

“…[10, p. 120]). As a consequence, the Einstein [8,20] G 2 /U (2)(U (2) represented by the short root) [8,20] [8,20] E 8 /SO (14) × U (1) s = 2 1 − =2 [8,20] SU [27,29] G 2 /U (2) (U (2) represented by the long root) [27] (1) SU (n)/T (n ≥ 4) s = n(n − 1)/2 ≥ n!/2 + 1 + E 1 [6] SU (2m)/T (m ≥ 2) s = m(2m − 1) ≥ (2m)!/2 + 1 + E 1 + E 2 [34] SU (2m)/T (m ≥ 6) s = m(2m − 1) ≥ (2m)!/2 + 1 + E 1 + E 2 + E 3 [21] SU (2m + 1)/T (m ≥ 6) s = m(2m + 1) ≥ (2m + 1)!/2 + 1 + E 1 + E 2 + E 3 SO(5)/T s = 4 -≥ 6 [34] SO(2n + 1)/T (n ≥ 12)…”

Equigeodesics on Generalized Flag Manifolds with Two Isotropy Summands

“…According to [2] and [3], a generalized flag manifolds has two isotropy summands if, and only if, Π K = Π − {α i0 } such that the simple root has Table 1. Generalized flag manifolds with two isotropy summands…”

“…In particular all vectors in these subspaces are structural equigeodesic vectors. 3 , α 4 α 5 , α 6 } be a system of simple roots for E 6 such that the highest root is given by μ = α 1 + 2α 2 + 2α 3 +3α 4 +2α 5 +α 6 . The flag manifold E 6 /SU (5)×SU (2)×U (1) is determined by Π K = Π − {α 5 }.…”