On Existence and Uniqueness Theorems for a Geometric Initial Value Problem in Local Isometric Embedding

In this paper a local problem of isometric embedding is studied. Here a sufficiant regular curve the initial curve is complemented to a surface of a prescribed line element an isometric embedding. While the local problem of isometric embedding is already in the focus of scientific research, solutions with a prescribed initial curve have not been considered by any other author this far. Therefore, tools are developed that are appropriate to deal with this problem, i.e. the geometric invariants of a curve, of a normal cut of a curve and of a surface and their connection. Further, the number of solutions is also discussed here.