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Invariant Kähler-Einstein metrics on compact homogeneous spaces. (English. Russian original) Zbl 0641.53050
Funct. Anal. Appl. 20, 171-182 (1986); translation from Funkts. Anal. Prilozh. 20, No. 3, 1-16 (1986).
The problem of describing homogeneous Einstein metrics presents great interest from the point of view of geometry, as well as of physical applications. Metrics of Riemannian symmetric spaces and of isotropically irreducible spaces, and Bergman metrics of homogeneous bounded regions are well-known examples of such metrics. Other examples of homogeneous Einstein metrics are known. The present work is dedicated to the explicit description of homogeneous Kähler-Einstein metrics on compact manifolds. From the results [J. L. Koszul, Can. J. Math. 7, 562–576 (1955; Zbl 0066.16104); J. Hano, Am. J. Math. 79, 885–900 (1957; Zbl 0096.16203), and Y. Matsushima, Nagoya Math. J. 46, 161–173 (1972; Zbl 0249.53050)] it follows that these metrics exhaust all nonplanar nonsymmetric Kähler-Einstein metrics admitting a transitive group of isometries, as well as all Kähler-Einstein metrics on compact manifolds admitting a transitive group of holomorphic transformations.

MSC:
53C30 Differential geometry of homogeneous manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:
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