Lipschitz minimality of Hopf fibrations and Hopf vector fields

DeTurck, Dennis¹ ; Gluck, Herman¹ ; Storm, Peter¹

¹Department of Mathematics, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33 Street, Philadelphia, PA 19104-6395, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 3, pp. 1369-1412

Résumé

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.

Détail

DOI : 10.2140/agt.2013.13.1369

Classification : 53C23, 53C30, 55R10, 55R25, 57R22, 57R25, 57R35, 53C38, 53C43
Keywords: Riemannian submersion, Lipschitz constant, Lipschitz minimizer, Hopf fibration, Hopf vector field, Grassmannian

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