Hermitian metrics on Calabi--Eckmann manifolds. (Q1397835)

The main purpose of the paper is to provide explicit examples of Riemannian metrics and their associated tensors with respect to a given almost complex structure. To achieve this, the authors consider the Calabi-Eckmann manifolds \(S^{2n+1}\times S^{2m+1}\) and by using the Hopf fibration perform the calculations with respect to a family of Riemannian submersions \(S^{2n+1}\times S^{2m+1}\rightarrow \mathbb{C} P^n\times \mathbb{C} P^m\), parametrized by a function \(\phi\) on the base. They compute explicitly the Ricci tensor, scalar curvature, J-Ricci tensor and J-scalar curvature of each metric of the family. Finally, they use the results to construct the behaviour of the Hermitian quantities versus those that are purely Riemannian.