The existence of invariant Einstein metrics on a compact homogeneous space (Q2837777)

Let \(G/H\) be a homogeneous space of a compact Lie group \(G\). We denote by \({\mathcal M}_1^G\) the Riemannian manifold of all \(G\)-invariant Riemannian metrics of volume 1 on \(G/H\). If a metric from \({\mathcal M}_1^G\) is Einstein, it is the critical point of the function \({\mathcal M}_1^G\rightarrow{\mathbb R}\), \(g\mapsto {\mathrm sc}(g)\). Let further \({\mathcal S}\) denote the topological space of geodesic rays from a point. NEWLINENEWLINENEWLINEIn a previous work, C. Böhm constructed a subspace \(X_\varepsilon\subset{\mathcal S}\) and a polyhedron \(\|{\mathcal K}\|\subset X_\varepsilon\). The vertices of \(\|{\mathcal K}\|\) are subalgebras of \({\mathfrak{g}}\) and the simplexes are flags of subalgebras. Using this polyhedron, he formulated a criterion on the existence of invariant positively definite Einstein metric on \(G/H\). NEWLINENEWLINENEWLINEIn another work by C. Böhm, M. Wang and W. Ziller, a related criterion was obtained and is referred to as graph theorem. NEWLINENEWLINENEWLINEIn the present paper, the two previous works are revisited. The space \(X_\varepsilon\) is constructed in a different way using the family \(\mathcal J\) of toral \(H\)-subalgebras and a subset \(\|{\mathcal J}\|\) of symmetric linear operators. As a subset of \({\mathfrak{gl}}({\mathfrak{g}})\), it holds \(\|{\mathcal K}\|\cap \|{\mathcal J}\|=\emptyset\). Some constructions from the previous works are clarified and refined. An alternative version of the Graph Theorem is formulated and a family of homogeneous spaces \(G/H\) on which there exist invariant Einstein metrics are constructed. These examples cannot be derived from older versions of criterions.