Geometry of Kaluza–Klein metrics on the sphere $${\mathbb{S}^3}$$

Abstract

In this paper, we introduce a new family of Riemannian metrics \({\tilde g_{\lambda\mu\nu}}\) on the three-sphere and study its geometric properties, starting from the description of their curvature. Such metrics, which include the standard metric g0 and Berger metrics on \({\mathbb{S}^3}\) as special cases, are called “of Kaluza–Klein type”, because they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle \({T_1 \mathbb{S}^2(\kappa)}\). Each sphere \({(\mathbb{S}^3, \tilde g_{\lambda\mu\nu})}\) is a homogeneous space, and we obtain a full classification of its homogeneous structures. Moreover, we introduce and study a natural almost contact structure \({(\varphi,\xi,\eta)}\), for which \({(\varphi,\xi,\eta,\tilde g_{\lambda\mu\nu})}\) is a (homogeneous) almost contact metric structure on the three-sphere. Finally, we see that for a suitable family of Kaluza–Klein type metrics \({\tilde g_{ac}}\) on \({\mathbb{S}^3}\), it is possible to construct a two-parameter family of harmonic morphisms from \({(\mathbb{S}^3, \tilde g_{ac})}\) to \({\mathbb{S}^2(\kappa)}\).