Some geometric characterizations of the Hopf fibration of the three-sphere

SALVAI, MARCOS

MONATSHEFETE FUR MATHEMATIK

SPRINGER WIEN

Año: 2006 vol. 147 p. 173 - 177

0026-9255

The Fréchet manifold E/~ of all embeddings(up to orientation preserving reparametrizations) of the circle in S^3 has a canonical weak Riemannian metric. We use the characterization obtained by H. Gluck and F. Warner of the oriented great circle fibrations of S^3 to prove that among all such fibrations p : S^3 --> B, the manifold B consisting of the oriented fibers is totally geodesic in E/~, or has minimum volume or diameter with the induced metric, exactly when p is a Hopf fibration.