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In this paper, we describe a new surprising example of a fibration of theClifford torus S3 x S3 in the 7-sphere by great 3-spheres, which is fiberwisehomogeneous but whose fibers are not parallel to one another. In particular itis not part of a Hopf fibration.  A fibration is fiberwise homogeneous when for any two fibers there is anisometry of the total space taking fibers to fibers and taking the first givenfiber to the second one.  We also describe in detail the geometry of this surprising fibration and howit differs from the Hopf fibration.

A surprising fibration of S3 x S3 by great 3-spheres