Sub-Riemannian geodesics and heat operator on odd dimensional spheres

Abstract

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.

24 pages, submitted

References in corpus (1)

• Sub-Riemannian geometry of parallelizable spheres

Cited by in corpus (2)

• Hopf fibration: geodesics and distances
• A Hamiltonian Approach to the Heat Kernel of a SubLaplacian on S(2n+1)