Alexey Podobryaev

We consider left-invariant optimal control problems on connected Lie groups. We describe the symmetries of the exponential map that are induced by the symmetries of the vertical part of the Hamiltonian system of the Pontryagin maximum principle. These symmetries play a key role in investigation of optimality of extremal trajectories. For connected Lie groups such that the generic coadjoint orbit has codimension not more than 1 and a connected stabilizer we introduce a general construction for such symmetries of the exponential map.

We give an example of a Riemannian manifold homeomorphic to a sphere such that its diameter cannot be realized as a distance between antipodal points. We consider a Berger sphere, i.e., a three-dimensional sphere with Riemannian metric that is compressed along the fibers of the Hopf fibration. We give a condition for a Berger sphere to have the desired property. We use our previous results on a cut locus of Berger spheres obtained by the method from geometric control theory.