Standard homogeneous Einstein manifolds and Diophantine equations
Yu. G. Nikonorov, Eugene D. Rodionov · Czech digital mathematics library · 1996
. Some new examples of standard homogeneous Einstein manifolds with semisimple transitive groups of motions and semisimple isotropy subgroups are constructed. For the construction of these examples the solutions of some systems of Diophantine equations are used. Let g and h be the Lie algebras of the compact connected Lie groups G and H, and let g be semisimple, g = g 1 \\Phi ::: \\Phi g r , where g 1 ; :::; g r are simple Lie algebras. We put B(X;Y ) = \\Gammatr(adXadY ) for all X;Y 2 g, and we define the standard Riemannian metric ae B on G=H as the metric obtained from B(X;Y ) under the projection ß : G ! G=H. We note that in [1]-[6] a classification was given of the simply connected compact standard homogeneous Einstein manifolds (G=H; ae B ) either with a simple transitive group of motions G, or with a simple isotropy subgroup H. Moreover, in the case of semisimple Lie groups G and H we have constructed new examples of standard homogeneous Einstein manifolds in the following way [...