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On the Ricci curvature of homogeneous metrics on noncompact homogeneous spaces. (English. Russian original) Zbl 0947.53025

Sib. Math. J. 41, No. 2, 349-356 (2000); translation from Sib. Mat. Zh. 41, No. 2, 421-429 (2000).
The author considers noncompact homogeneous spaces \(G/H\) with noncompact semisimple Lie group \(G\) and compact Lie group \(H\). Let \(K\) be a maximal compact subgroup of \(G\). We have the Cartan decomposition \(g=k \oplus p'\) and the orthogonal decomposition with respect to the Killing form \(k = h \oplus p''\). The author proves that there are no Einstein homogeneous metrics on \(G/H\) which provide orthogonality of the modules \(p'\) and \(p''\). In particular, \(SO(a+b, c+d)/ SO(a) \times SO(b) \times SO(c) \times SO(d)\) and \(Sp(a+b, c+d)/ Sp(a) \times Sp(b) \times Sp(c) \times Sp(d)\) have no homogeneous Einstein metrics.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
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References:

[1] Helgason S., Differential Geometry and Symmetric, Spaces [Russian translation], Mir, Moscow (1964). · Zbl 0122.39901
[2] Besse A. L., Einstein Manifolds [Russian translation], Mir, Moscow (1990). · Zbl 0734.53004
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