## Algebraic & Geometric Topology

### Lipschitz minimality of Hopf fibrations and Hopf vector fields

#### 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
Revised: 12 October 2012
Accepted: 22 October 2012
First available in Project Euclid: 19 December 2017

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

#### 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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