Hermitian metrics on Calabi-Eckmann manifolds

We consider a family of Riemannian submersions S2n+1 X S2m+1 --> CPn x CPn parametrized by a function phi on the base, whose squared exponential is used as a dilation factor on the fibers. The total space of these submersions is endowed with the complex structure of Calabi-Eckmann, and each member of the phi-family of metrics is Hermitian relative to this structure. We compute explicitly the Ricci tensor, scalar curvature, J-Ricci tensor and J-scalar curvature of each of these metrics, and use the results to contrast the behaviour of the Hermitian quantities versus those that are purely Riemannian. The fibers of these submersions may be collapsed or blown-up with these tensors showing significant differences as this takes place. We show that among metrics in this family, the only ones that have constant J-scalar curvature are those corresponding to phi equal to a constant, and distinguish further these metrics by analyzing their behaviour relative to a suitable family of deformations. (C) 2002 Elsevier Science B.V. All rights reserved.