On conformal submersions with geodesic or minimal fibers
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On conformal submersions with geodesic or minimal fibers Tomasz Zawadzki1 Received: 9 January 2020 / Accepted: 3 June 2020 / Published online: 24 June 2020 © The Author(s) 2020
Abstract We prove that every conformal submersion from a round sphere onto an Einstein manifold with fibers being geodesics is—up to an isometry—the Hopf fibration composed with a conformal diffeomorphism of the complex projective space of appropriate dimension. We also show that there are no conformal submersions with minimal fibers between manifolds satisfying certain curvature assumptions. Keywords Conformal submersions · Foliations · Einstein metrics Mathematics Subject Classification 53C12 · 53C21 · 53C20
1 Introduction One of the common problems of Riemannian geometry and theory of foliations is the existence of foliations and distributions, satisfying certain geometric properties, on a given Riemannian manifold. Examining the natural representation of the product of orthogonal groups yields several interesting classes of distributions [14], described by their extrinsic geometry: totally geodesic, totally umbilical, and minimal. When the orthogonal distribution to the foliation belongs to one of these classes, we call the foliation Riemannian, conformal, or an SL(q)-foliation (respectively), where q is the dimension of the distribution. Of special interest are those foliations for which both distributions, tangent and orthogonal to the leaves, belong to one (not necessarily the same) of the families in the above classification. The aim of this paper is to study one such case—of a conformal foliation with minimal fibers, with a particular focus on conformal fibrations of spheres (here always considered “round”, i.e., equipped with the standard Riemannian metric induced from Euclidean space) by great circles. Fibrations of spheres by great circles have been thoroughly examined from the topological and differential point of view [6, 7]. It was established that they exist only on spheres of odd dimension, the leaf space of such fibration is diffeomorphic to the complex projective space of appropriate dimension [12], and that any such fibration deformation retracts * Tomasz Zawadzki [email protected] 1
Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90‑238 Lodz, Poland
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Annals of Global Analysis and Geometry (2020) 58:191–205
to the Hopf fibration [15]. Fibrations of spheres, satisfying additional geometric assumption: giving rise to a Riemannian foliation by their fibers, were examined in [4, 18, 22]. A generalization of this assumption—when the fibration defines a conformal foliation, and its one-dimensional fibers are not necessarily great circles—was examined on the threedimensional sphere in [10]. Due to these works, Riemannian submersions from spheres and conformal submersions from the 3-sphere are fully classified from the point of view of Riemannian geometry. They all can be related to the Hopf fibration by a pair of isometries (in Riemannian foliation
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