Trivializable sub-Riemannian structures on spheres. (English) Zbl 1288.53025
A manifold \((M, {\mathcal D}, g)\) is called a sub-Riemannian manifold if it has a distribution \({\mathcal D}\) which is bracket generating, that is, all linear combinations of vector fields of \(T{\mathcal D}\) of \(TM\) and a finite number of their Lie brackets span the tangent bundle \(TM\). Then the sub-Riemannian structure on \(M\) is called trivializable (briefly TSR) if \({\mathcal D}\) is spanned by a family of globally defined vector fields on \(M\). In this paper, the authors classify the TSR-structures on spheres \(S^7\) and \(S^{15}\) induced by a Clifford module structure of \(\mathbb R^8\) and \(\mathbb R^{16}\), respectively. Also, they show that TSR-structures only exist on \(S^N\) \((N = 3, 7, 15)\). The rest of the paper includes properties of such TSR-structures, followed by some open problems in this area.
Reviewer: Krishan Lal Duggal (Windsor)
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