Some Geometric Characterizations of the Hopf Fibrations of the Three-Sphere

The Fréchet manifold 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 π:S ³→B, the manifold B consisting of the oriented fibers is totally geodesic in , or has minimum volume or diameter with the induced metric, exactly when π is a Hopf fibration. Partially supported by foncyt, Antorchas, ciem (conicet) and secyt (unc).