On small Carnot-Carathéodory spheres (Q1578266)

A Carnot-Carathéodory metric on a manifold \(M\) is a positive definite metric tensor defined on a smooth distribution \(\Delta\subset TM\). The distance of two points \(p,q\in M\) induced by such a structure is the infimum of the length of all \(\Delta\)-horizontal curves joining \(p\) and \(q\). In this paper the author proves that sufficiently small spheres in a Carnot-Carathéodory metric are indeed homeomorphic to Euclidean spheres, provided that one of the two following conditions is satisfied: (a) \(\Delta\) admits a one-parameter group of contractions that leave it invariant; (b) \(\Delta\) is of length 2, i.e., \([\Delta,\Delta]=TM\).