U(n + 1) × U(p + 1)-Hermitian metrics on the manifold S2n+1 × S2p+1

A two-parameter family of invariant almost-complex structures Jα,c is given on the homogeneous space M × M’ = U(n + 1)/U(n) × U(p + 1)/U(p); all these structures are integrable. We consider all invariant Riemannian metrics on the homogeneous space M × M’. They depend on five parameters and are Hermitian with respect to some complex structure Jα,c. In this paper, we calculate the Ricci tensor, scalar curvature, and obtain estimates of the sectional curvature for any metric on M × M’. All the invariant metrics of nonnegative curvature are described. We obtain the extremal values of the scalar curvature functional on the four-parameter family of metrics gα,c,λ,λ’;1.