×

Singularity analysis of Lorentzian hypersurfaces on pseudo \(n\)-spheres. (English) Zbl 1329.53034

Summary: This paper considers the one parameter families of extrinsic differential geometries of Lorentzian hypersurfaces on pseudo \(n\)-spheres. We give a continuous relationship among the three types of Gauss indicatrices by a one parameter map. Meanwhile, we also give the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo \(n\)-spheres by the Legendrian singularity theory.

MSC:

53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53A35 Non-Euclidean differential geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Duggal, Foliations of lightlike hypersurfaces and their physical interpretation, Central European Journal of Mathematics 10 pp 1789– (2012) · Zbl 1259.53017 · doi:10.2478/s11533-012-0067-x
[2] Hiscock, Models of evaporating black holes. II. Effects of the outgoing created radiation, Physical Review D 23 pp 2823– (1981) · doi:10.1103/PhysRevD.23.2823
[3] Semi-Riemannian Geometry with Applications to Relativity (1983) · Zbl 0531.53051
[4] Little, On singularities of submanifolds of higher dimensional Euclidean space, Annali di Matematica Pura ed Applicata 83 pp 261– (1969) · Zbl 0187.18903 · doi:10.1007/BF02411172
[5] Pei, Singularities of lightlike hypersurface in semi-Euclidean 4-space with index 2, Science in China, Series A 53 (12) pp 3243– (2010) · Zbl 1223.58032 · doi:10.1007/s11425-010-4121-5
[6] Pei, The Lorentzian surfaces in semi-Euclidean 4 space with index 2, Journal of Mathematical Analysis and Applications 385 (1) pp 243– (2012) · Zbl 1227.53021 · doi:10.1016/j.jmaa.2011.06.037
[7] Sun, Families of Gauss indicatrices on Lorentzian hypersurfaces in pseudo-spheres in semi-Euclidean 4 space, Journal of Mathematical Analysis and Applications 400 (1) pp 133– (2013) · Zbl 1259.53019 · doi:10.1016/j.jmaa.2012.10.043
[8] Sun, Null surfaces of null curves on 3-null cone, Physics Letters A 378 (14-15) pp 1010– (2014) · Zbl 1296.53043 · doi:10.1016/j.physleta.2014.02.002
[9] Tari, Families of Gauss indicatrices on smooth surfaces in pseudo-spheres in the Minkowski 4-space, Journal of Singularities 1 pp 116– (2010) · Zbl 1293.53014 · doi:10.5427/jsing.2010.1h
[10] Wang, Singularities of focal surfaces of null Cartan curves in Minkowski 3-space, Abstract and Applied Analysis 2012 (Article ID 8238) pp 20 pages– (2012)
[11] Whitney, On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane, Annals of Mathematics 62 pp 374– (1955) · Zbl 0068.37101 · doi:10.2307/1970070
[12] Zakalyukin, Lagrangian and Legendre singularities, Functional Analysis and Its Applications 10 pp 26– (1976) · Zbl 0331.58007 · doi:10.1007/BF01075768
[13] Ervin, An improved boundary element method for the charge density of a thin electrified plate in R3, Mathematical Methods in the Applied Sciences 13 (4) pp 291– (1990) · Zbl 0717.65093 · doi:10.1002/mma.1670130403
[14] Montaldi, On generic composites of maps, Bulletin of the London Mathematical Society 23 pp 81– (1991) · Zbl 0733.58001 · doi:10.1112/blms/23.1.81
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.