Sub-Riemannian geodesics and heat operator on odd dimensional spheres

Abstract

In this article we study the sub-Riemannian geometry of the spheres S2n+1 and S4n+3, arising from the principal S1-bundle structure defined by the Hopf map and the principal S3-bundle structure given by the quaternionic Hopf map, respectively. The S1 action leads to the classical contact geometry of S2n+1, while the S3 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 S7 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.