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

作者：Li Chengbo; Zhan Huaying^*

来源：Journal of Geometry and Physics, 2013, 65: 1-6.

DOI：10.1016/j.geomphys.2012.11.008

• Sub-Riemannian structures
• Complex Hopf fibrations
• Jacobi equations
• Conjugate points
• Comparison theorems
• Magnetic field on Riemannian manifolds

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 [2].

• 出版日期2013-3
• 单位天津理工大学