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.
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 
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/
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     title = {Einstein metrics on a class of five-dimensional  homogeneous spaces},
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     pages = {389--393},
     year = {1991},
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     zbl = {0747.53037},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1991_32_2_a21/}
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[K-V] Kowalski O., Vanhecke L.: Classification of five-dimensional naturally reductive spaces. Math. Proc. Camb. Phil. Soc. 97 (1985), 445-463. | MR | Zbl