On $K-P$ sub-Riemannian Problems and their Cut Locus

The problem of finding minimizing geodesics for a manifold M with a sub-Riemannian structure is equivalent to the time optimal control of a driftless system on M with a bound on the control. We consider here a class of sub-Riemannian problems on the classical Lie groups G where the dynamical equations are of the form · x=ΣjXj(x)uj and the Xj=Xj(x) are right invariant vector fields on G and uj:=uj(t) the controls. The vector fields Xj are assumed to belong to the P part of a Cartan K-P decomposition. These types of problems admit a group of symmetries K which act on G by conjugation. Under the assumption that the minimal isotropy group in K is discrete, we prove that we can reduce the problem to a Riemannian problem on the regular part of the associated quotient space G/K. On this part we define the corresponding quotient metric. For the special cases of the K-P decomposition of SU(n) of type AIII we prove that the assumption on the minimal isotropy group is verified. As an example of application of the techniques discussed we find the cut locus of a K-P optimal control problem on SU(2).