Geodesic fibrations of elliptic 3-manifolds

The well-known Hopf fibration of S3 is interesting in part because its fibers are geodesics, or great circles, of S3. However, this is not the only great circle fibration of S3. In 1983, Herman Gluck and Frank Warner used the fact that the space of all oriented geodesics of the 3-sphere is homeomorphic to S2 × S2 to establish that there are many other great circle fibrations of S3. They showed that a submanifold of S2 × S2 corresponds to a fibration of S3 by oriented great circles if and only if it is the graph of a distance decreasing map from either S2 factor to the other. Since S3 is the universal cover of all elliptic 3-manifolds, we use this result to investigate geodesic Seifert fibrations of elliptic 3-manifolds. We also develop a different perspective on the space of oriented geodesics in S3 than that used by Gluck and Warner, and we examine its role in studying the geometry of the 3-sphere