We construct the curvature-type differential invariants for a wide class of geometric structures and controlsystems on manifolds, especially for sub-Riemannian structures on nonholonomic vector distributions. We then can calculate the curvature maps for a class of sub-Riemannian structures on distributions having additional transversal infinitesimal symmetry and investigate the comparison theorems for the number of conjugate points along the sub-Riemannian extremals. Finally, we review the construction of the curvature maps for three dimensional contact sub-Riemannian case and give a new proof which could possibly to be applied to the higher dimensions. T he talk is mainly based on the joint works with Igor Zelenko.

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