$U(n+1)\times U(p+1)$-Hermitian Metrics on the Manifold $S^{2n+1}\times S^{2p+1}$

A two-parameter family of invariant almost-complex structures $J_{a,c}$ is given on the homogeneous space $M\times M'=U(n+1)/U(n)\times U(p+1)/U(p)$; all these structures are integrable. We consider all invariant Riemannian metrics on the homogeneous space $M\times M'$. They depend on five parameters and are Hermitian with respect to some complex structure $J_{a,c}$. In this paper, we calculate the Ricci tensor, scalar curvature, and obtain estimates of the sectional curvature for any metric on $M\times M'$. 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 $g_{a,c,\lambda,\lambda';1}$.