On existence of invariant Einstein metrics on a compact homogeneous space
Graev, Michail M.
Original · EN
We prove that the existence of a positively defined, invariant Einstein metric m on a connected homogeneous space G/H of a compact Lie group G is the consequence of non-contractibility of some compact set C=XG,ₕΣ (Böhm polyhedron) introduced by C.Böhm. There is a natural continuous map of C onto the flag complex KB of a finite graph B. The special case of C = KB, KB non-contractible, is one of Böhm existence criteria, and the case of the graph B non-connected is a improved version of the Graph Theorem (C.Böhm, M.Wang, and W.Ziller) actual for any z(g). Moreover, preparation theorems of C. Böhm on retractions are revisited and new constructions of some topologic spaces are suggested.
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