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.
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Godoy Molina, M., Markina, I. Sub-Riemannian geodesics and heat operator on odd dimensional spheres. Anal.Math.Phys. 2, 123–147 (2012). https://doi.org/10.1007/s13324-012-0028-3
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DOI: https://doi.org/10.1007/s13324-012-0028-3