Narrow Results

We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvature and for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails.

We construct a form of the swallowtail singularity in $R^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtails.

Modeling the worldsheets along the trajectory (a string) of a point particle (a 0-brane) as two timelike surfaces in 3-subspace of Minkowskian spacetime, respectively, the work aims at analyzing the topological structures of these two worldsheets along the trajectory of the point particle from the view point of singularity theory. Different from the regular curves, the traveling trajectory (modeled as framed timelike curve) of the particle is allowed to be singular, two worldsheets are generated by the traveling trajectories of the particle. As applications of singularity theory, we classify the singularities of these two worldsheets along the traveling trajectory of the particle. Using the approach of the unfolding theory in singularity theory, we find two new geometric invariants which are useful for characterizing the local topological structures of singularities of these two worldsheets along the particle. It is revealed that there exist cuspidal edge type and swallowtail type of singularities for these two worldsheets under the appropriate conditions of geometric invariants. Meanwhile, it is also pointed out that the types of singularities of these two worldsheets have a close relationships to the order of contacts between these two worldsheets and two timelike planes, respectively. Finally, some examples are presented to interpret our theoretical results.

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 ${\mathrm{\Delta }}_1$-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 ${\sigma }_d(s)$, depending on whether ${\sigma }_d(s)\ne 0$ or ${\sigma }_d(s)=0$ and ${\sigma }_d^{\prime }(s)\ne 0$, the singularities of these worldsheets can be characterized by the geometric invariant. We provide two explicit examples of worldsheets to illustrate the theoretical results.

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.

In this paper, we investigate the special properties of geometrical particles with null paths in de Sitter 3-space–time, new Frenet equations and an important invariant associated with null paths are presented. By means of unfolding theory, the local topological structure of the lightlike dual surfaces is revealed. It is found that the lightlike dual surface has some singularities whose types can be determined by the invariant. Based on the theory of Legendrian dualities on pseudospheres and the theory of contact manifolds, it is shown that there exists the ${\mathrm{\Delta }}_4$-dual relationship between the lightlike transversal trajectory of the particle and the lightlike dual surface. In addition, an interesting and important fact mentioned is that the contact of lightlike transversal trajectory with lightcone quadric and the contact of lightlike transversal trajectory with null hyperplane have the same order when they are related to the same type of singularities of the lightlike dual surface.

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 ${\mathrm{\Delta }}_1$-dual each other.

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.

In this paper, as applications of singularity theory, we consider the singularities of a world sheet and the traveling trajectory (i.e. a worldline) of a geometric particle when they are confined in a world hyper-sheet and are generated by a Sabban framed base curve in de Sitter 3-space. Using the approach of the unfolding theory in singularity theory, we establish the relationships between the world sheet, the worldline of the geometric particle and two invariants to characterize the local topological structures of singularities of the world sheet and the worldline of the geometric particle via these two invariants, respectively. Moreover, we also describe the dual relationships between the tangent curve of framed base curve and the worldline of the geometric particle. An important fact indicates that the worldline $L_{\gamma }$ and the tangent curve $t$ of the original curve $\gamma$ are ${\mathrm{\Delta }}_5$-dual each other. Finally, two concrete examples are presented to interpret our theoretical results.

In this work, we model two event horizons associated with a null Cartan curve as two lightlike hypersurfaces, respectively. We define a lightlike surface and a spacelike surface whose images coincide with the sets of critical values of two event horizons, and meanwhile we present two curves whose images coincide with the sets of critical values of these two surfaces, respectively. Using the singularity theory, we characterize the local topological structures of two event horizons, two surfaces and two curves at their singularities by means of two new invariants. Moreover, we also present a spacelike braneworld model along the particle as a spacelike surface in hyperbolic 3-space. An important fact shows that from the viewpoint of Legendrian dualities, this surface is ${\mathrm{\Delta }}_2$-dual to the tangent trajectory $L(t)$ of the null Cartan curve in Lorentz–Minkowski space-time. Meanwhile, we also consider a curve whose image is the set of critical values of this surface in hyperbolic 3-space. The third invariant of the null Cartan curve characterizes the singularities of the surface $\mathcal{ℒ}\mathcal{𝒮}$ and the curve $\mathcal{𝒫}$ in hyperbolic 3-space. A result indicates that surface $\mathcal{ℒ}\mathcal{𝒮}$ is locally diffeomorphic to the swallowtail $SW$ or cuspidal edge $CE$ and $\mathcal{𝒫}$ is locally diffeomorphic to the $(2,3,4)$-cusp at certain a singular point. It is also shown that there exist deep relationships between the singularities of the surface $\mathcal{ℒ}\mathcal{𝒮}$ and the curve $\mathcal{𝒫}$ and the order of contact between $L(t)$ and elliptic quadric ${ℰ}{𝒬}(v_0,-1)$ or the order of contact between $L(t)$ and spacelike hyperplane $\mathrm{\text{HP}}(v_0,-1)$. Finally, we present several examples to describe the main results.

For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with fixed boundary conditions and prescribed singular pattern, by which we mean the combinatorial information of how the 1-dimensional singular locus meets the boundary. Each step of our algorithm modifies the given map only locally by either creating or eliminating a pair of cusps. We show that the number of cusps is an invariant modulo 2 and can be reduced to at most one, and we compute the minimal number of components of the singular locus in terms of the prescribed data. Applications include a discussion of pseudo-immersions as well as the computation of state sums in Banagl’s positive topological field theory.