The sub-Riemmanian geometry of three-dimensional Berger spheres

Abstract

We study the integrability of the sub-Riemannian geodesic flow induced by the general left-invariant Riemannian metric on 𝑆3. We are interested in two different classes of this problem: the first is associated with the left-invariant distribution and the second with the right-invariant distribution. It is well known that the standard metric on the sphere 𝑆 3 is bi-invariant and the corresponding geodesic flow is integrable in the non-commutative sense for both types of distributions. Not surprisingly, the same statement holds for the arbitrary left-invariant metric associated with the left-invariant distribution. The Berger spheres form a special class of examples of left-invariant metrics obtained from the standard metric by shrinking along the fibers of a Hopf fibration. We show that the Hamiltonian LR system corresponding to the left-invariant Berger metric and the right-invariant distribution is integrable in the ommutative sense.