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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 structuredefined by the Hopf map and the principal $S^3-$bundle structure given by thequaternionic Hopf map respectively. The $S^1$ action leads to the classicalcontact geometry of $S^{2n+1}$, while the $S^3$ action gives another type ofsub-Riemannian structure, with a distribution of corank 3. In both cases themetric is given as the restriction of the usual Riemannian metric on therespective horizontal distributions. For the contact $S^7$ case, we give anexplicit form of the intrinsic sub-Laplacian and obtain a commutation relationbetween the sub-Riemannian heat operator and the heat operator in the verticaldirection.

Sub-Riemannian geodesics and heat operator on odd dimensional spheres