Hopf fibration: geodesics and distances
Der-Chen Chang, Irina Markina, and Alexander Vasil'ev
TL;DR
This paper investigates geodesics on odd-dimensional spheres respecting the Hopf fibration, providing complete solutions for the 3-sphere, partial results for higher dimensions, and calculating the Carnot-Carathéodory distance, with motivations from quantum mechanics.
Contribution
It offers a complete solution for geodesics on the 3-sphere respecting the Hopf fibration and partial results for higher dimensions, along with explicit distance calculations.
Findings
Complete solution for geodesics on 3-sphere
Partial results for higher-dimensional spheres
Explicit Carnot-Carathéodory distance calculation
Abstract
Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carath\'eodory distance is calculated. We also present some motivations related to quantum mechanics.
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