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Given a Hopf fibration of a round sphere by parallel great subspheres, weprove that the projection map to the base space is, up to isometries of domainand 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 fibres as a cross-section of theunit tangent bundle of the sphere, and prove that it is, up to isometries ofdomain and range, the unique Lipschitz constant minimizer in its homotopyclass. Previous attempts to find a mathematical sense in which Hopf fibrationsand Hopf vector fields are optimal have met with limited success.

Lipschitz minimality of Hopf fibrations and Hopf vector fields