Recognition of singularities of rectifying worldsheets generated by framed worldlines

In this paper, we consider the local topological structures of a class of new worldsheets, call it the rectifying worldsheets, which are generated by a class of singular worldlines. Using the classification approaches of the finite type on the tangent developables and defining the extended striction curve, this paper gives the detailed classification of the rectifying worldsheets of singular worldlines. It is demonstrated that the rectifying worldsheets of singular worldlines will appear not only in cuspidal edge and swallowtail, but cuspidal beaks under suitable conditions. Especially the singularities of rectifying worldsheets of singular worldlines are associated with curvature functions such that the singularities can be characterized by these functions. Two examples are provided to put the theoretical results into the practice of computation and classification.

Hyperbolic worldsheets and worldlines of null Cartan curves in de Sitter 3-space

The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are [Formula: see text]-dual each other.

Cuspidal edges with the same first fundamental forms along a knot

Letting [Formula: see text] be a compact [Formula: see text]-curve embedded in the Euclidean [Formula: see text]-space ([Formula: see text] means real analyticity), we consider a [Formula: see text]-cuspidal edge [Formula: see text] along [Formula: see text]. When [Formula: see text] is non-closed, in the authors’ previous works, the local existence of three distinct cuspidal edges along [Formula: see text] whose first fundamental forms coincide with that of [Formula: see text] was shown, under a certain reasonable assumption on [Formula: see text]. In this paper, if [Formula: see text] is closed, that is, [Formula: see text] is a knot, we show that there exist infinitely many cuspidal edges along [Formula: see text] having the same first fundamental form as that of [Formula: see text] such that their images are non-congruent to each other, in general.

Singularities of worldsheets associated with null Cartan curves in Lorentz–Minkowski space–time

In this paper, as applications of singularity theory, we study the singularities of several worldsheets generated by null Cartan curves in Lorentz–Minkowski space–time. Using the approach of the unfolding theory in singularity theory, we establish the relationships between these worldsheets and invariants such that the cuspidal edge type of singularity and the swallowtail type of singularity can be characterized by these invariants, respectively. Meanwhile, the contact of the tangent curve of a null Cartan curve with some model surfaces are discussed in detail. In addition, we also describe the dual relationships between the tangent curve of a null Cartan curve and these worldsheets. Finally, some concrete examples are provided to explain our theoretical results.

Singularities of worldsheets in spherical space–times

In this paper, the singularities of the geometry for four classes of worldsheets, which are respectively, located in three-dimensional hyperbolic space and three-dimensional de Sitter space–time are considered. Under the theoretical frame of geometry of space–time and as applications of singularity theory, it is shown that these worldsheets have two classes of singularities, that is, in the local sense, these four classes of worldsheets are, respectively, diffeomorphic to the cuspidal edge and the swallowtail. The first hyperbolic worldsheet and the second hyperbolic worldsheet are [Formula: see text]-dual to the tangent curves of spacelike curves. Moreover, it is also revealed that there is a close relationship between the types of singularities of worldsheets and a geometric invariant [Formula: see text], depending on whether [Formula: see text] or [Formula: see text] and [Formula: see text], the singularities of these worldsheets can be characterized by the geometric invariant. We provide two explicit examples of worldsheets to illustrate the theoretical results.

Mod2 local invariants of maps between 3-manifolds

This paper refers to the work [V. Goryunov, Local invariants of maps between 3-manifolds, J. Topology 6 (2013) 757–776] on local invariants of maps between 3-manifolds. It is assumed that the manifolds have no boundary, and that the source is compact. In the case when the source and the target are oriented, Goryunov proved that every local order one invariant with integer values can be written as a linear combination of seven basic invariants, and gave a geometrical interpretation for them. When the target is the oriented [Formula: see text], there are further four basic mod2 invariants. One of the mod2 invariants has been provided with a topological interpretation, in terms of the number of components and of the self-linking of a framed link constructed from the cuspidal edge. Here, we show that two further independent linear combinations of the mod2 invariants have a topological interpretation, involving the self-linking number of two curves defined by all irregular points of the critical value set of a generic map from an oriented closed 3-manifold to [Formula: see text].

Geometric Invariants of Cuspidal Edges

We give a normal form of the cuspidal edge that uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges that determine them up to order three. We also clarify relations between these invariants.

COUNTING SINGULARITIES VIA FITTING IDEALS

The stable singularities of differential map germs constitute the main source of studying the geometric and topological behavior of these maps. In particular, one interesting problem is to find formulae which allow us to count the isolated stable singularities which appear in the discriminant of a stable deformation of a finitely determined map germ. Mond and Pellikaan showed how the Fitting ideals are related to such singularities and obtain a formula to count the number of ordinary triple points in map germs from ℂ2 to ℂ3, in terms of the Fitting ideals associated with the discriminant. In this article we consider map germs from (ℂn+m, 0) to (ℂm, 0), and obtain results to count the number of isolated singularities by means of the dimension of some associated algebras to the Fitting ideals. First in Corollary 4.5 we provide a way to compute the total sum of these singularities. In Proposition 4.9, for m = 3 we show how to compute the number of ordinary triple points. In Corollary 4.10 and with f of co-rank one, we show a way to compute the number of points formed by the intersection between a germ of a cuspidal edge and a germ of a plane. Furthermore, we show in some examples how to calculate the number of isolated singularities using these results.

Design of developable surfaces and the application of thin-walled developable structures

This manuscript is an attempt to collect and systematize all cardinal scientific results of geometrical design of nondegenerate developable surfaces with a cuspidal edge. Information on the application of thin-walled developable structures and developable surfaces has also been presented. Wide choices of design methods of developable surfaces provide not only necessary shapes and special properties but they also prove to be convenient to apply. This surface is actively applied for design of ship hulls, in agricultural machine building, in aircraft construction, in pipe design for making the diverse transitions, in road building, in cartography and in civil engineering.