A Class Of Calabi-Yau Manifolds Constructed By The Hopf Fibration

In General Relativity,the solution of the Einstein vacuum field equations with vanishing cosmological spatial constant is a class of Ricci-flat Lorentzian metrics,Taub-NUT metrics are special solutions of such equations,and then Gibbons and Hawking derive the correspond-ed Ricci-flat metrics in the Riemannian case.Such metrics are simultaneously Calabi-Yau metrics with special symmetry.In this paper,we will illuminate that Gibbons-Hawking metrics can generate from doubly warped products on R4×S1 by hopf fibration,the doubly warped products have the form of maetrics?²?r?(ds_R4²+?²?r?d?²)We will derive the formulas of doubly warped metrics' curvatures calculations on I x S3 x S1,and also present the formulas of the warped metrics and their conformal metrics which is constructed by the Hopf fibration of doubly warped metrics with a perspctive of Riemannian submersions.And then we apply this perspetive into Gibbons-Hawking metrics and try to generalize this construction.We will discuss how to construct more example of Calabi-Yau manifolds by this method at the last part of this paper.