  The present paper starts with an introduction to quaternions and then defines
the 3-dimmensional sphere as the set of quaternions of length one. The
quaternion group induces on $\mathbb{S}^3$ a structure of noncommutative Lie
group. This group is compact and the results obtained in this case are very
different than those obtained in the case of the Heisenberg group, which is a
noncompact Lie group. Like in the Heisenberg group case, we introduce a
nonintegrable distribution on the sphere and a metric on it using two of the
noncommutative left invariant vector fields. This way $\mathbb{S}^3$ becomes a
subRiemannian manifold. It is known that the group $SU(2) $ is isomorphic with
the sphere $\mathbb{S}^3$ and represents an example of subRiemannian manifold
where the elements are matrices. The main issue here is to study the
connectivity by horizontal curves and its geodesics on this manifold. In this
paper, we are using Lagrangian method to study the connectivity theorem on
${\mathbb S}^3$ by horizontal curves with minimal arc-length. We show that for
any two points in ${\mathbb S}^3$, there exists such a geodesic joining these
two points.
