Algebraic & Geometric Topology

Lipschitz minimality of Hopf fibrations and Hopf vector fields

Dennis DeTurck, Herman Gluck, and Peter Storm

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715587

Digital Object Identifier
doi:10.2140/agt.2013.13.1369

Mathematical Reviews number (MathSciNet)
MR3071130

Zentralblatt MATH identifier
1268.53054

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.agt/1513715587


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