Trivializable sub-Riemannian structures on spheres

We classify the trivializable sub-Riemannian structures on odd-dimensional spheres SN that are induced by a Clifford module structure of RN+1. The underlying bracket generating distribution is of step two and spanned by a set of global linear vector fields X1,…,Xm. As a result we show that such structures only exist in the cases where N=3,7,15. The corresponding hypo-elliptic sub-Laplacians Δsub are defined as the (negative) sum of squares of the vector fields Xj. In the case of a trivializable rank four distribution on S7 and a trivializable rank eight distribution on S15 we obtain a part of the spectrum of Δsub. We also remark that in both cases there is a relation between the eigenfunctions and Jacobi polynomials.