Curvature and the equivalence problem in sub-Riemannian geometry

These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms connections, frame bundles and geometry. Then we arrive main aim these notes, which is description canonical grading connection existing on sub-Riemann manifolds with constant symbol. structures are exactly what needed order determine if two isometric. three concrete examples, Engel (2,3,4)-manifolds, contact Cartan (2,3,5)-manifolds. edited version a lecture series given at 42nd Winter school: Geometry Physics, Srní, Czech Republic, mostly based [8] other earlier work. However, work (2,3,4)-manifolds original research, illustrate important special case were our model has minimal set isometries.