On lightlike hypersurfaces and lightlike focal sets of null Cartan curves in Lorentz-Minkowski spacetime

Abstract

In this paper, as a type of event horizons in astrophysics, a class of lightlike hypersurfaces that is generated by null curves will be investigated and discussed. Based on discussions of the properties of the local differential geometry of null curves and singularity theory, we provide classifications of the singularities of lightlike hypersurfaces and lightlike focal sets. In addition, we reveal the facts that the types of these singularities and the order of contact between a null Cartan curve and a pseudosphere are related closely to null Cartan curvatures. Finally, examples of lightlike hypersurfaces and lightlike focal set are used to demonstrate our theoretical results.

Lightlike hypersurfaces and lightlike focal sets with respect to Bishop frame in 4-dimensional Minkowski Space E1-4.

“…Then, in 1998, Akivis et al investigate the singular points of light-like hypersurfaces of the de Sitter space S n+1 [1]. The singularities of light-like surface and hypersurfaces have been studied in [22,23].…”

Lightlike surfaces of Darboux‐like indicatrixes and binormal indicatrixes of spacelike curves in lightcone 3‐space

“…14 In the last few years, much more information on the topic have become available. [15][16][17][18][19][20][21][22][23][24][25][26] When ones consider the problem of the space curves lying in a nondegenerate spacetime model, for example, de Sitter space, anti de Sitter space, or hyperbolic space, the top priority is to find a suitable frame on such curves, without exception, the contributors adopted the frames in which vectors are pseudo-orthogonal each other and there do not exist lightlike vectors; unfortunately, due to the degeneracy of lightcone, such a frame is not suitable when we consider a space curve lying in lightcone and require the position vector of the curve to contain in the frame. Serve as the need, some attempts have been made to discuss the space curves lying in nullcone of index two in our previous study.…”

“…We draw the projection (in purple) of the image of the lightlike surface  + of binormal indicatrix and its critical value set  + (s, 0) (in green) to 3-space (see Figures 16,17,18,19). To watch the locus, the critical value set of  + , clearly, we separate the images of critical value set of  + from the images of the lightlike surface  + (see Figures 20,21,22,23). In each of Figures 16-23, the red part corresponds to the cusipidal edge type of singularity and the green one corresponds to the swallowtail type of singularity.…”

Singularities of Indefinite Hypersurfaces and Indefinite Surfaces along Timelike Curves in Nullcone with Index Two

“…L. Kong and D. Pei [8] investigated the singularities of lightlike surfaces and focal surfaces of spacelike curves in hyperbolic spacetime sphere, and some geometry properties of the spacelike curves are obtained. The second author and his collaborators presented abundant research results concerning the singularities of submanifolds in semiEuclidean space [12][13][14][15]. In [13], they investigated the null developables of timelike curves that lie on nullcone in 3-dimensional semi-Euclidean space with index 2 and classified the singularities of the null developables of timelike curves.…”