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Lipschitz minimality of
Hopf fibrations and Hopf vector fields
Dennis DeTurck, Herman Gluck and Peter Storm |
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Algebraic & Geometric Topology 13 (2013) 1369–1412
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Abstract |
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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. |
Keywords
Riemannian submersion, Lipschitz constant, Lipschitz
minimizer, Hopf fibration, Hopf vector field, Grassmannian
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Mathematical Subject Classification 2010
Primary: 53C23, 53C30, 55R10, 55R25, 57R22, 57R25, 57R35
Secondary: 53C38, 53C43
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References |
Publication
Received: 23 May 2012
Revised: 12 October 2012
Accepted: 22 October 2012
Published: 24 April 2013
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Authors |
| Dennis DeTurck | |
| Department of Mathematics University of Pennsylvania David Rittenhouse Laboratory 209 South 33 Street Philadelphia, PA 19104-6395 USA |
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| Herman Gluck | |
| Department of Mathematics University of Pennsylvania David Rittenhouse Laboratory 209 South 33 Street Philadelphia, PA 19104-6395 USA |
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| Peter Storm | |
| Department of Mathematics University of Pennsylvania David Rittenhouse Laboratory 209 South 33 Street Philadelphia, PA 19104-6395 USA |
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