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On small Carnot-Carathéodory spheres. (English) Zbl 0967.53021

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., \([\Delta,\Delta]=TM\).

MSC:

53C17 Sub-Riemannian geometry
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