  Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation)
$\eta,\bar\eta$ of the velocities in these points. In this paper we are
interested to the problem of minimizing the cost $$ J[\gamma]=\int_0^T
g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))+
  K^2_{\gamma(t)}g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) ~dt$$ along all
smooth curves starting from $x$ with direction $\eta$ and ending in $\bar{x}$
with direction $\bar\eta$. Here $g$ is the standard Riemannian metric on $S^2$
and $K_\gamma$ is the corresponding geodesic curvature.
  The interest of this problem comes from mechanics and geometry of vision. It
can be formulated as a sub-Riemannian problem on the lens space L(4,1).
  We compute the global solution for this problem: an interesting feature is
that some optimal geodesics present cusps. The cut locus is a stratification
with non trivial topology.
