Vector fields on bifurcation diagrams of quasi singularities

Fawaz Alharbi; Yanlin Li; Fawaz Alharbi; Yanlin Li

doi:10.3934/math.20241710

AIMS Mathematics

2024, Volume 9, Issue 12: 36047-36068. doi: 10.3934/math.20241710

Previous Article Next Article

Research article

Vector fields on bifurcation diagrams of quasi singularities

Fawaz Alharbi ¹,
Yanlin Li ^{2
,
,}

1.
Department of Mathematics, College of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia
2.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China

Received: 03 October 2024 Revised: 11 December 2024 Accepted: 13 December 2024 Published: 26 December 2024
MSC : 57R45, 53A05

We describe the generators of the vector fields tangent to the bifurcation diagrams and caustics of simple quasi boundary singularities. As an application, submersions on the pair $ (G, B) $, which consists of a cuspidal edge $ G $ in $ \mathbb{R}^3 $ that contains a distinguishing regular curve $ B $, are classified. This classification was used as a means to investigate the contact that a general cuspidal edge $ G $ equipped with a regular curve $ B\subset G $ has with planes. The singularities of the height functions on $ (G, B) $ are discussed and they are related to the curvatures and torsions of the distinguished curves on the cuspidal edge. In addition to this, the discriminants of the versal deformations of the submersions that were accomplished are described and they are related to the duality of the cuspidal edge.
- bifurcation diagram,
- caustic,
- vector field,
- cuspidal edge,
- contact,
- curvatures and torsions,
- height function,
- contact,
- deformations,
- discriminat,
- daul
Citation: Fawaz Alharbi, Yanlin Li. Vector fields on bifurcation diagrams of quasi singularities[J]. AIMS Mathematics, 2024, 9(12): 36047-36068. doi: 10.3934/math.20241710

Related Papers:

Abstract

We describe the generators of the vector fields tangent to the bifurcation diagrams and caustics of simple quasi boundary singularities. As an application, submersions on the pair $ (G, B) $, which consists of a cuspidal edge $ G $ in $ \mathbb{R}^3 $ that contains a distinguishing regular curve $ B $, are classified. This classification was used as a means to investigate the contact that a general cuspidal edge $ G $ equipped with a regular curve $ B\subset G $ has with planes. The singularities of the height functions on $ (G, B) $ are discussed and they are related to the curvatures and torsions of the distinguished curves on the cuspidal edge. In addition to this, the discriminants of the versal deformations of the submersions that were accomplished are described and they are related to the duality of the cuspidal edge.

References

[1]	V. M. Zakalyukin, Reconstructions of fronts and caustics depending on a parameter and versality of mappings, J. Math. Sci., 27 (1984), 2713–2735. https://doi.org/10.1007/BF01084818 doi: 10.1007/BF01084818
[2]	V. Arnold, Wave front evolution and equivariant Morse lemma, Commun. Pur. Appl. Math., 29 (1976), 557–582. https://doi.org/10.1002/cpa.3160290603 doi: 10.1002/cpa.3160290603
[3]	J. W. Bruce, Vector fields on discriminants and bifurcation varieties, Bull. London Math. Soc., 17 (1985), 257–262. https://doi.org/10.1112/blms/17.3.257 doi: 10.1112/blms/17.3.257
[4]	A. Alghanemi, A. Alghawazi, The $\lambda$-point map between two Legendre plane curves, Mathematics, 11 (2023), 977–997. https://doi.org/10.3390/math11040997 doi: 10.3390/math11040997
[5]	T. Fukui, M. Hasegawa, Singularities of parallel surfaces, Tohoku Math. J., 64 (2012), 387–408. https://doi.org/10.2748/tmj/1347369369 doi: 10.2748/tmj/1347369369
[6]	Y. Li, E. Guler, Right conoids demonstrating a Time-like axis within minkowski Four-Dimensional space, Mathematics, 12 (2024), 2421. https://doi.org/10.3390/math12152421 doi: 10.3390/math12152421
[7]	Y. Li, H. Abdel-Aziz, H. Serry, F. El-Adawy, M. Saad, Geometric visualization of evolved ruled surfaces via alternative frame in Lorentz-Minkowski 3-space, AIMS Math., 9 (2024), 25619–25635. https://doi.org/10.3934/math.20241251 doi: 10.3934/math.20241251
[8]	Y. Li N. Turki, S. Deshmukh, O. Belova, Euclidean hypersurfaces isometric to spheres, AIMS Math., 9 (2024), 28306–28319. https://doi.org/10.3934/math.20241373 doi: 10.3934/math.20241373
[9]	Y. Li, M. S. Siddesha, H. A. Kumara, M. M. Praveena, Characterization of Bach and Cotton Tensors on a Class of Lorentzian Manifolds, Mathematics, 12 (2024), 3130. https://doi.org/10.3390/math12193130 doi: 10.3390/math12193130
[10]	Y. Li, S. Bhattacharyya, S. Azami, Li-Yau type estimation of a semilinear parabolic system along geometric flow, J. Inequal Appl., 131 (2024). https://doi.org/10.1186/s13660-024-03209-y doi: 10.1186/s13660-024-03209-y
[11]	Y. Li, A. K. Mallick, A. Bhattacharyya, M. S. Stankovic, A conformal $\eta$-Ricci soliton on a Four-Dimensional lorentzian Para-Sasakian manifold, Axioms, 13 (2024), 753. https://doi.org/10.3390/axioms13110753 doi: 10.3390/axioms13110753
[12]	F. Alharbi, V. Zakalyukin, Quasi corner singularities, P. Steklov I. Math., 270 (2010), 1–14. https://doi.org/10.1134/S0081543810030016 doi: 10.1134/S0081543810030016
[13]	F. Alharbi, Quasi cusp singularities, J. Sing., 12 (2015), 1–18. https://doi.org/10.5427/jsing.2015.12a doi: 10.5427/jsing.2015.12a
[14]	F. Alharbi, S. Alsaeed, Quasi semi-border singularities, Mathematics, 7 (2019), 495. https://doi.org/10.3390/math7060495 doi: 10.3390/math7060495
[15]	F. Alharbi, Bifurcation diagrams and caustics of simple quasi border singularities, Topo. Appl., 9 (2012), 381–388. https://doi.org/10.1016/j.topol.2011.09.011 doi: 10.1016/j.topol.2011.09.011
[16]	J. W. Bruce, J. M. West, Functions on cross-caps, Math. Proc. Cambridge, 123 (1988), 19–39.
[17]	A. P. Francisco, Functions on a swallowtail, arXiv Prep., 53 (2023), 52–74. https://doi.org/10.48550/arXiv.1804.09664 doi: 10.48550/arXiv.1804.09664
[18]	R. O. Sinha, F. Tari, On the geometry of the cuspidal edge, Osaka J. Math., 55 (2018), 393–421.
[19]	R. O. Sinha, K. Saji, On the geometry of folded cuspidal edges, Rev. Mat. Complut., 31 (2018), 627–650. https://doi.org/10.1007/s13163-018-0257-6 doi: 10.1007/s13163-018-0257-6
[20]	J. Damon, A-equivalence and the equivalence of sections of images and discriminants, Singular. Theory Appl., 1462 (1991), 93–121. https://doi.org/10.1007/BFb0086377 doi: 10.1007/BFb0086377
[21]	D. Mond, R. Buchweitz, Linear free divisors and quiver representations, London Math. Soc. Lecture Note Ser., 324 (2005), 18–20.
[22]	V. Arnold, Singularities of caustics and wave fronts, Dordrecht: Kluwer Academic Publishers, 1990.
[23]	J. W. Bruce, P. J. Giblin, Curves and singularities: A geometrical introduction to singularity theory, Cambridge University Press, 1984.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)