2005/01/28 by A. Raouf Chouikha, Chouikha, A. Raouf
Mathematics · Medicine · Physics and Astronomy · #53C21 #53C25 #58G30 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #Spaceflight effects on biology #math.AP #math.DG #msc:53C21 #msc:53C25 #msc:58G30
paper · pdf · doi:10.48550/arxiv.math/0501505
43 pages
arxiv created 2005/01/28 · openalex publication_date 2005/01/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class [g0] of the Riemannian product S1× Sn-1 : a circle of length T crossed with the (n-1)-dimensional standard sphere. Such metrics have a harmonic Riemannian curvature and a non parallel Ricci tensor, except for the cylindric one. Thus, it appears a natural link between them and the Derdzinski metrics which are warped product and classify a family of Riemannian manifolds. These two families actually differ by conformal transformations. Moreover, we are interested in the multiplicity problem of the pseudo-cylindric metrics in [g0]. We also study the existence problem and the number of the Derdzinski metrics. Furthermore, we prove that the pseudo-cylindric metrics may be expressed in terms of elliptic functions for the dimension n = 3,4 and 6 only, and in terms of automorphic functions for any other dimension. This fact allows us to give new bounds for global estimates. Finally, we examine the curvature of the asymptotic pseudo-cylindric metrics which are complete singular Yamabe metrics on the standard sphere punctured of k points. We show that any of such (non trivial) metric has a non parallel Ricci tensor.