A co-dimensional 3 sub-Riemannian structure on Gromoll-Meyer exotic sphere

Abstract

We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a $4n+3$ dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere $\mathbb{S}^7$.