Generalized Cheeger-Gromoll Metrics and the Hopf map

TL;DR

This paper constructs a family of Riemannian metrics on the tangent bundle of a two-sphere that induce constant curvature metrics on the unit tangent bundle, linking the Hopf map to Riemannian submersions, with a hyperbolic analogue also discussed.

Contribution

It introduces new Riemannian metrics on tangent bundles of spheres and hyperbolic planes that produce constant curvature structures and relate to the Hopf map as a Riemannian submersion.

Findings

Existence of metrics inducing constant curvature on unit tangent bundles

Identification of the Hopf map as a Riemannian submersion

Extension to hyperbolic tangent bundles

Abstract

We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented.