A note on sub-Riemannian structures associated with complex Hopf fibrations

TL;DR

This paper investigates the curvature properties of sub-Riemannian structures on odd-dimensional spheres linked to the Hopf fibration, connecting them with the Fubini-Study metric and estimating conjugate points.

Contribution

It expresses curvature maps of these structures using known Riemannian and curvature form data, providing a framework for comparison theorems in sub-Riemannian geometry.

Findings

Curvature maps expressed via Fubini-Study tensor and Hopf fibration curvature form

Estimation of conjugate points based on curvature bounds

Application to general corank 1 sub-Riemannian structures with symmetries

Abstract

Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature form of the Hopf fibration. We also estimate the number of conjugate points of a sub-Riemannian extremal in terms of the bounds of the sectional curvature and the curvature form. It presents a typical example for the study of curvature maps and comparison theorms for a general corank 1 sub-Riemannian structure with symmetries done by C.Li and I.Zelenko.