Geodesic
Einstein metrics on a class of five-dimensional homogeneous spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 389-393.

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We prove that there is exactly one homothety class of invariant Einstein metrics in each space $SU(2) \times SU(2) / SO(2)_r (r\in Q, \, |r|\neq 1)$ defined below.
Classification : 53C25, 53C30
Keywords: homogeneous Riemannian manifolds; Einstein manifolds; Ricci tensor; sectional curvature 
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     title = {Einstein metrics on a class of five-dimensional  homogeneous spaces},
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Rodionov, E. D. Einstein metrics on a class of five-dimensional  homogeneous spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 389-393. http://geodesic.mathdoc.fr/item/CMUC_1991__32_2_a21/