Functions on a crosscap

The study of the differential geometry of surfaces in 3-space has a long and celebrated history. Over the last 20 years a new approach using techniques from singularity theory has yielded some interesting results (see, for example [3, 5, 19] for surveys). Of course surfaces arise in a number of ways: they are often defined explicitly as the image of a mapping f: R2[rightward arrow]R3. Since the subject is differential geometry one normally asks that these defining mappings are smooth, that is infinitely differentiable, however it is not true, in any sense, that most such parametrisations will yield manifolds. For such mappings have self-intersections, and more significantly they may have crosscaps (also known as Whitney umbrellas). Moreover if we perturb the maps these singularities will persist; that is they are stable. (See [14, 20] for details.) Consequently when studying the differential geometry of surfaces in 3-space there are good reasons for studying surfaces with crosscaps. In this paper we carry out a classification of mappings from 3-space to lines, up to changes of co-ordinates in the source preserving a crosscap. We can apply our results to the geometry of generic crosscap points. In [10] we computed geometric normal forms for the crosscap and used them to study the dual of the crosscap. We shall see that the approach here yields more information than that obtained in [10], although the latter has the advantage of being more explicit (in particular various aspects of the geometry can be compared using the normal forms). We refer the reader to [16, 18] for background material concerning singularity theory.