Cone Metrics Near Characteristic Points in Sub-Riemannian Geometry

This thesis studies the local metric geometry of hypersurfaces in sub-Riemannian manifolds in a neighbourhood of characteristic points. Their blow-up and the resulting approximating geometries are investigated, with a focus on the emergence of conic metric structures. A characterization of the quadratic hypersurfaces admitting a cone structure is presented in the setting of Heisenberg groups.