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

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• 作者：Chengbo Li<sup>asup><sup> ; sup><sup>chengboli@tju.edu.cnsup> ; Huaying Zhan<sup>bsup><sup> ; sup><sup>zhanhuaying@gmail.comsup>
• 关键词：53C17 ; 70G45 ; 49J15 ; 34C10
• 刊名：Journal of Geometry and Physics
• 出版年：2013
• 出版时间：March, 2013
• 年：2013
• 卷：65
• 期：Complete
• 页码：1-6
• 全文大小：218 K

文摘

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 theorems for a general corank 1 sub-Riemannian structure with symmetries done by C.?Li and I.?Zelenko (2011) in?.