New homogeneous Einstein metrics on \(\mathrm{SO}(7)/T\)
Generalized flag manifolds are homogeneous manifolds \(M=G/C(T)\), where \(C(T)\) is the centralizer of a torus \(T\) in \(G\). In the case \(C(T)=T\), we have the full flag manifold \(M=G/T\). In simple cases of full flag manifolds, invariant Einstein metrics on them were described in previous works. In the present paper, invariant Einstein metrics on the full flag manifold \(M=\mathrm{SO}(7)/T\) are studied. This manifold has nine isotropy summands and the Einstein equations are rather complicated. The 48 solutions are determined numerically and it is shown that only 5 of them are mutually non-isometric. One Einstein metric is Kähler and four of them are non-Kähler.
scientific article; zbMATH DE number 6838916