Homogeneous Einstein metrics on flag manifolds
Let \(K\) be a connected compact semisimple Lie group, \(T\) a maximal torus of \(K\) and \(M = K/T\) a flag manifold. It is known that \(M\) admits a Kähler Einstein metric [\textit{A. Borel} and \textit{F. Hirzebruch}, Am. J. Math. 80, 458-538 (1958; Zbl 0097.36401)]. In this paper the author studies \(K\)-invariant Einstein metrics on \(M\) which are not Kähler. This problem was previously studied by \textit{D. V. Alekseevsky} [Differential geometry and its applications. Proc. Conf., Brno/Czech. 1986, Commun., 1-21 (1987; Zbl 0641.53051)] and \textit{A. Arvanitoyeorgos} [Trans. Am. Math. Soc. 337, No. 2, 981-995 (1993; Zbl 0781.53037)] in the more general context of generalized flag manifolds or Kähler \(C\)-spaces, constructing new \(K\)-invariant Einstein metrics on some generalized flag manifolds. The author gives an explicit expression for the Ricci tensor of \(M\) in the case that \(K\) is a classical simple Lie group using the notation introduced by [\textit{M. Wang} and \textit{W. Ziller}, Invent. Math. 84, 177-194 (1986; Zbl 0596.53040)]. In order to prove that there exist \(K\)-invariant Einstein metrics he computes a Gröbner basis for a system of polynomials of multi-variables and shows the existence of positive solutions for a system of algebraic equations.
scientific article; zbMATH DE number 1413875