Singular curves and invariants of geometric structures

Given a submanifold $V$ of the tangnet bundle to a smooth manifold $M$, we consider “admissible curves” on $M$ whose velocities belong to V. Among examples are parametrized by the length curves on a Riemannian manifold and integral curves of a vector distribution. The boundary map sends a curve into its endpoints. “Singular curves” are critical points of the boundary map restricted to the space of admissible curves. They give nice and efficient tools for the investigation and classification of many intersting geometric structures.