## 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

### Keywords:

homogeneous space; Einstein manifold
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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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