  In this article we study the sub-Riemannian geometry of the spheres
$S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure
defined by the Hopf map and the principal $S^3-$bundle structure given by the
quaternionic Hopf map respectively. The $S^1$ action leads to the classical
contact geometry of $S^{2n+1}$, while the $S^3$ action gives another type of
sub-Riemannian structure, with a distribution of corank 3. In both cases the
metric is given as the restriction of the usual Riemannian metric on the
respective horizontal distributions. For the contact $S^7$ case, we give an
explicit form of the intrinsic sub-Laplacian and obtain a commutation relation
between the sub-Riemannian heat operator and the heat operator in the vertical
direction.
