Abstract. In Carnot groups of step≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.
For a H-type group G, we first give explicit equations for its shortest sub-Riemannian geodesies. We use properties of sub-Riemannian geodesies in G to characterise the isometry group ISO(G) with respect to the Carnot-Caratheodory metric. It turns out that ISO(G) coincides with the isometry group with respect to the standard Riemannian metric of G.
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the sub-Riemannian structure and is "symmetric" and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.
The notion of horizontal energy minimizers between C-C spaces is introduced. We prove existence of such energy minimizers when the domain is a C 2 , noncharacteristic bounded open set in a C-C space and the target is a C-C space of Carnot type.
In Carnot groups of step≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.
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