The Hopf fibrations are characterized by being fiberwise homogeneous

The Hopf fibrations of S2n+1 by great circles, S4n+3 by great 3-spheres, and S15 by great 7-spheres are the prototypical examples of fibrations of spheres by subspheres, and have a number of features that make them attractive. For example, they have parallel fibers, and what\u27s more, they are the only fibrations of spheres by subspheres which have this property. They also have a great deal of symmetry. They are fiberwise homogeneous : for any two fibers, there is an isometry of the total space taking the first given fiber to the second, and preserving fibers. The main result of this dissertation is that the Hopf fibrations are characterized by this property among all fibrations of round spheres by subspheres. That is, they deserve to be promoted to our attention for being so symmetric. We develop an intuition for the property of fiberwise homogeneity by classifying all the fibrations of Euclidean and Hyperbolic 3-space by geodesics having this feature. They are analogues of the Hopf fibration of the 3-sphere by great circles. We also find new nonstandard examples of fiberwise homogeneous fibrations of the Clifford Torus S3 × S3 in the 7-sphere by great 3-spheres which are not restrictions of the Hopf fibration. Finally, we prove a local version of our main result: that open sets in the 3-sphere which are fibered by great circles and are locally fiberwise homogeneous are subsets of the Hopf fibration