Sub-Riemannian geodesics and heat operator on odd dimensional spheres

In this article we study the sub-Riemannian geometry of the spheres S2n+1 and S4n+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 S2n+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.