Local differential geometry of cuspidal edge and swallowtail

Abstract

We investigate the local differential geometric invariants of cuspidal edge and swallowtail from the view point of singularity theory. We introduce finite type invariants of such singularities (see Remark 1.5 and Theorem 2.11) based on certain normal forms for cuspidal edge and swallowtail. Then we discuss several geometric aspects based on our normal form. We also present several asymptotic formulas concerning our invariants with respect to Gauss curvature and mean curvature. Typical examples of wave fronts are parallel surfaces of a regular surface in the 3dimensional Euclidean space, and it is well-known that such surfaces may have several singularities like cuspidal edge and swallowtail. Singularity types of parallel surfaces are investigated in [3], and the next interest is to investigate local differential geometries of such singularities. There are several attempts to describe them. For instance, K. Saji, M. Umehara, and K. Yamada ([12]) defined the notion of singular curvature κs and normal curvature κν of cuspidal edge, and, later, K. Saji and L. Martins ([7]) described all invariants up to order 3. It is clear that there are more differential geometric invariants in higher order terms, and to describe all such invariants up to finite order is one motivation of the paper. Since Gauss curvature and mean curvature are often diverge at singularities and we are interested in their asymptotic behaviors near a singularity in terms of our invariants. We are going to describe their asymptotic behaviors of our local differential geometric invariants of cuspidal edge near swallowtail. An ideas of singularity theory is to reduce a given map-germ (R2, 0) → (R3, 0) to certain normal form (see [9], for example). Their normal forms are obtained up to -equivalence where  is the group of coordinate changes of the source and the target. In that context, we reduce a given map-germ to one of normal forms in the list there, composing certain coordinate changes of the source and the target. For differential geometric purpose, general coordinate changes of the target are too rough, since they do not preserve differential geometric properties, and we should restrict the coordinate change of the target to the motion group. From this point, we will consider the product group of coordinate change of the source with the motion group of the target (the rotation group when we consider map-germs) and we introduce a normal form for cuspidal edge (see (1.1)) and swallowtail (Theorem 2.4) by the equivalence relation defined by this group. We believe that this is a powerful method to investigate singular surfaces, since this unable us to describe all differential geometric 2010 Mathematics Subject Classification. Primary 57R45; Secondary 53A05. Dedicated to Professor Takashi Nishimura on the occasion of his 60th birthday.