Some properties of the metric on Calabi-Eckmann manifolds
[Objective]Although Kaehler manifolds have been extensively studied,non-Kaehler manifolds have not been.Calabi-Eckmann manifolds were introduced by Calabi and Eckmann,and their complex structures and related properties were first studied.In recent years,numerous studies on complex submanifolds,cohomology,and deformation on Calabi-Eckmann manifolds have emerged.In this article,we study some properties of metrics on Calabi-Eckmann manifolds.Corresponding results of Hopf manifolds are extended to Calabi-Eckmann manifolds,and the fact that it is a quotient space of its universal covering spaceCn\{0} under the action of its fundamental group is used in the study of Hopf manifolds.Unfortunately,this method cannot be extended to Calabi-Eckmann manifolds because it is a simply connected non-Kaehler manifold.[Methods]Herein,we use the Calabi-Eckmann manifold with the form of S2m+1×S2n+1,which can be considered as a complex analytic fiber bundle with elliptic curves S1 X S1 as fibers over CPm X CPn.Then,we construct a globally defined(1,1)Kaehler form on the base space manifold CPm X CPn,obtain a globally defined volume form,and finally construct a Kaehler form on the Calabi-Eckmann manifold pullded back from a Kaehler form on CPm XCPn.[Results]We prove that the holomorphic immersions of the Calabi-Eckmann manifold on its base space manifold are not ddc-exact,and thus the Calabi-Eckmann manifold is non-pluriclosed.Additionally,we prove that for a Kaehler form ω on the Calabi-Eckmann manifold,ddcω≤0,and thus establish that the Calabi-Eckmann manifold is plurinegative.[Conclusion]Some properties of the metric of Calabi-Eckmann manifolds deserve further investigations.Hopefully,the proposed global Kaehler form can be used to study whether the induced metric is balanced and 1-symmetric.
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