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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Lipschitz minimality of Hopf fibrations and Hopf vector fields

Dennis DeTurck, Herman Gluck and Peter Storm

Algebraic & Geometric Topology 13 (2013) 1369–1412
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.

Keywords
Riemannian submersion, Lipschitz constant, Lipschitz minimizer, Hopf fibration, Hopf vector field, Grassmannian
Mathematical Subject Classification 2010
Primary: 53C23, 53C30, 55R10, 55R25, 57R22, 57R25, 57R35
Secondary: 53C38, 53C43
References
Publication
Received: 23 May 2012
Revised: 12 October 2012
Accepted: 22 October 2012
Published: 24 April 2013
Authors
Dennis DeTurck
Department of Mathematics
University of Pennsylvania
David Rittenhouse Laboratory
209 South 33 Street
Philadelphia, PA 19104-6395
USA
Herman Gluck
Department of Mathematics
University of Pennsylvania
David Rittenhouse Laboratory
209 South 33 Street
Philadelphia, PA 19104-6395
USA
Peter Storm
Department of Mathematics
University of Pennsylvania
David Rittenhouse Laboratory
209 South 33 Street
Philadelphia, PA 19104-6395
USA