Algebraic & Geometric Topology

Lipschitz minimality of Hopf fibrations and Hopf vector fields

Dennis DeTurck, Herman Gluck, and Peter Storm

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Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class.

Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibers as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class.

Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success.

Article information

Algebr. Geom. Topol., Volume 13, Number 3 (2013), 1369-1412.

Received: 23 May 2012
Revised: 12 October 2012
Accepted: 22 October 2012
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 55R10: Fiber bundles 55R25: Sphere bundles and vector bundles 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx] 57R25: Vector fields, frame fields 57R35: Differentiable mappings
Secondary: 53C38: Calibrations and calibrated geometries 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

Riemannian submersion Lipschitz constant Lipschitz minimizer Hopf fibration Hopf vector field Grassmannian


DeTurck, Dennis; Gluck, Herman; Storm, Peter. Lipschitz minimality of Hopf fibrations and Hopf vector fields. Algebr. Geom. Topol. 13 (2013), no. 3, 1369--1412. doi:10.2140/agt.2013.13.1369.

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