Geodesic
$U(n+1)\times U(p+1)$-Hermitian Metrics on the Manifold $S^{2n+1}\times S^{2p+1}$
Matematičeskie zametki, Tome 82 (2007) no. 2, pp. 207-223.

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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 $g_{a,c,\lambda,\lambda';1}$.
Keywords: Hermitian metric on a homogenous space, Ricci tensor, sectional curvature, Hopf fibration, scalar curvature functional, holomorphic function, Lie algebra, Riemannian connection. 
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     title = {$U(n+1)\times U(p+1)${-Hermitian} {Metrics} on the {Manifold} $S^{2n+1}\times S^{2p+1}$},
     journal = {Matemati\v{c}eskie zametki},
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N. A. Daurtseva. $U(n+1)\times U(p+1)$-Hermitian Metrics on the Manifold $S^{2n+1}\times S^{2p+1}$. Matematičeskie zametki, Tome 82 (2007) no. 2, pp. 207-223. http://geodesic.mathdoc.fr/item/MZM_2007_82_2_a5/

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