Strong Sard conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3

Abstract In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic distribution of rank 2 manifold, investigate size set points that can be reached by singular horizontal paths starting from point and it has Hausdorff dimension at most 1. fact, provided lengths curves under consideration are bounded with respect to complete Riemannian metric, demonstrate such is semianalytic curve. As consequence, combining our techniques recent developments regularity minimizing geodesics, geodesics in manifolds always class $$C^1$$ C 1 , actually they outside finite points.