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.
Highlights
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
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 by great circles
We examine conformal submersions from round spheres of any odd dimension, with fibers being great circles, using direct methods of Riemannian geometry, introduced in [16], and applied to conformal submersions in [9]
Summary
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. 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. We prove that the only Einstein manifold, that can be the image of a conformal submersion from the sphere with fibers being great circles, is isometric to the complex projective space with the Fubini-Study metric In this case, the submersion is the composition of the Hopf fibration and a conformal diffeomorphism of the complex projective space. These restrictions apply in particular to manifolds with metrics of constant scalar curvature, and, can be expressed in terms of conformal classes of Riemannian metrics
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