  We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ provided by
the restriction of the Riemannian metric of curvature 1 to the plane
distribution orthogonal to the Hopf vector field. We compute the geodesics
associated to the Carnot-Carath\'eodory distance and we show that, depending on
their curvature, they are closed or dense subsets of a Clifford torus.
  We study area-stationary surfaces with or without a volume constraint in
$(\mathbb{S}^3,g_{h})$. By following the ideas and techniques in [RR] we
introduce a variational notion of mean curvature, characterize stationary
surfaces, and prove classification results for complete volume-preserving
area-stationary surfaces with non-empty singular set. We also use the behaviour
of the Carnot-Carath\'eodory geodesics and the ruling property of constant mean
curvature surfaces to show that the only $C^2$ compact, connected, embedded
surfaces in $(\mathbb{S}^3,g_{h})$ with empty singular set and constant mean
curvature $H$ such that $H/\sqrt{1+H^2}$ is an irrational number, are Clifford
tori. Finally we describe which are the complete rotationally invariant
surfaces with constant mean curvature in $(\mathbb{S}^3,g_{h})$.
