Sub-Riemannian geodesics and heat operator on odd dimensional spheres

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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.