Trivializable sub-Riemannian structures on spheres

We classify the trivializable sub-Riemannian structures on odd-dimensional spheres S-N 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 X-1, ... , X-m. As a result we show that such structures only exist in the cases where N = 3, 7, 15. The corresponding hypo-elliptic sub-Laplacians Delta(sub) are defined as the (negative) sum of squares of the vector fields X-j. In the case of a trivializable rank four distribution on S-7 and a trivializable rank eight distribution on S-15 we obtain a part of the spectrum of Delta(sub). We also remark that in both cases there is a relation between the eigenfunctions and Jacobi polynomials. (C) 2012 Elsevier Masson SAS. All rights reserved.