Regular homotopy and total curvature I: circle immersions into surfaces

An immersion of manifolds is a map with everywhere injective differential. Two immersions are regularly homotopic if there exists a continuous 1–parameter family of immersions connecting one to the other. The Smale–Hirsch h–principle [8, 4] says that the space of immersions M → N , dim(M) < dim(N) is homotopy equivalent to the space of injective bundle maps TM → TN . In contrast to differential topological properties, differential geometric properties of immersions do not in general satisfy h–principles, see [3, (A) on page 62]. In this paper and the sequel [2], we study some aspects of the differential geometry of immersions and regular homotopies in the most basic cases of codimension one immersions. We investigate whether or not it is possible to perform topological constructions while keeping control of certain geometric quantities.