Global Differential Geometry of 1-Resolvable C∞ Curves in the Plane

A C∞ nonimmersed curve in the Euclidean plane c: S1 → E2 is 1-resolvable if its lift to the orthonormal frame bundle c: S1 → F extends to an immersion. The objective of this paper is to relate the shape of the planar curve with the differential topology of its lift. Specifically, we derive inequalities relating geometric invariants of c with topological invariants of c. The corresponding equalities will identify the simplest 1-resolvable curves.