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Fine structure of reductive pseudo-Kählerian spaces. (English) Zbl 0739.53040

The structure of a homogeneous pseudo-Kählerian manifold \(M=G/H\) with a reductive group \(G\) of automorphisms is studied. It is proved that the manifold \(M\) decomposes into a direct product of pseudo-Kählerian manifolds \(G/H=G_ 0/\{1\}\times G_ 1/H_ 1\times \dots \times G_ r/H_ r\) where \(G_ 0\) is abelian and \(G_ i\), \(i>0\) are simple subgroups of \(G\) and \(G=G_ 0\times G_ 1\times \dots\times G_ r\). The following generalization of a theorem due to A. Borel is proved. A homogeneous space \(G/H\) of a connected semisimple Lie group \(G\) carries an invariant pseudo-Kählerian structure iff \(H\) is the centralizer of a torus in \(G\). A description of the invariant pseudo-Kählerian structures on a homogeneous space \(G/C(T)\), where \(G\) is a simple Lie group and \(C(T)\) is the centralizer of a torus, is given.

MSC:

53C30 Differential geometry of homogeneous manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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