Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T06:45:16.095Z Has data issue: false hasContentIssue false
  • English
  • Français

PROJECTIONS OF HYPERSURFACES IN ℝ4 WITH BOUNDARY TO PLANES

Published online by Cambridge University Press:  13 August 2013

LUCIANA F. MARTINS  and
ANA CLAUDIA NABARRO
LUCIANA F. MARTINS
Affiliation:
Departamento de Matemática, IBILCE-UNESP, R. Cristóvão Colombo, 2265, CEP 15054-000, São José do Rio Preto, SP, Brazil e-mail: lmartins@ibilce.unesp.br
ANA CLAUDIA NABARRO
Affiliation:
Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação-ICMC, Universidade de São Paulo–Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil e-mail: anaclana@icmc.usp.br
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study orthogonal projections of generic embedded hypersurfaces in ℝ4 with boundary to 2-spaces. Therefore, we classify simple map germs from ℝ3 to the plane of codimension less than or equal to 4 with the source containing a distinguished plane which is preserved by coordinate changes. We also go into some detail on their geometrical properties in order to recognize the cases of codimension less than or equal to 1.

Keywords

53A0758K0558K40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 1 , January 2014 , pp. 149 - 167
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Arnol'd, V. I., Guseĭn-Zade, S. M. and Varchenko, A. N., Singularities of differentiable maps, vol. I, The classification of critical points, caustics and wave fronts, Monographs in Mathematics, 82 (Birkhäuser Boston, MA, 1985).CrossRefGoogle Scholar
2.Bruce, J. W., Classifications in singularity theory and their applications, in New developments in singularity theory (Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 31 July–11 August 2000), NATO Science Series II: Mathematics, Physics and Chemistry, vol. 21 (Kluwer Academic Publishers, Netherlands, 2001), 3–33.Google Scholar
3.Bruce, J. W., du Plessis, A. A. and Wall, C. T. C., Determinacy and unipotency, Invent. Math. 88 (1987), 521–554.Google Scholar
4.Bruce, J. W. and Giblin, P. J., Projections of surfaces with boundary, Proc. Lond. Math. Soc. 60 (3) (1990), 392–416.CrossRefGoogle Scholar
5.Bruce, J. W., Kirk, N. P. and Plessis, A. A. du, Complete transversals and the classification of singularities, Nonlinearity 10 (1) (1997), 253–275.CrossRefGoogle Scholar
6.Bruce, J. W., and Nogueira, A. C., Surfaces in ℝ4 and duality, Quart. J. Math. (Oxford) 49 (2) (1998), 433–443.Google Scholar
7.Damon, J. N., The unfolding and determinacy theorems for subgroups of $\mathcal{A}$ and $\mathcal{K}$, Mem. Amer. Math. Soc. 50 (306) (1984), 1–88.Google Scholar
8.Giblin, P. J. and Tari, F., Perpendicular bisectors, duals and local symmetry in the plane. Proc. R. Soc. Edinb. Sect. A 125 (1995), 181–194.CrossRefGoogle Scholar
9.Kirk, N. P., Computational aspects of classifying singularities, LMS J. Comput. Math. 3 (2000), 207–228.CrossRefGoogle Scholar
10.Martins, L. F., Oliveira, R. D. S. and Tari, F., On pairs of regular foliations in ℝ3 and singularities of map-germs, Geom. Dedicata 135 (2008), 103–118.CrossRefGoogle Scholar
11.Mather, J. N., Stability of C∞ mappings, IV: Classification of stable germs by $\mathcal{R}$-algebras, Inst. Hautes Etudes Sci. Publ. Math. 37 (1969), 223–248.CrossRefGoogle Scholar
12.Mochida, D. K. H., Fuster, M. C. R. and Ruas, M. A. S., The geometry on surfaces in 4-space from a contact viewpoint, Geom. Dedicata 54 (1995), 323–332.CrossRefGoogle Scholar
13.Nabarro, A. C., Projections of hypersurfaces in ℝ4 to planes, Lecture Notes in Pure and Applied Maths, vol. 232 (Marcel Dekker, New York, 2003), 283–300.Google Scholar
14.Nabarro, A. C., Duality and contact of hypersurfaces in ℝ4 with hyperplanes and lines, Proc. Edinb. Math Soc. 46 (2003), 637–648.CrossRefGoogle Scholar
15.Nuño-Ballesteros, J. J. and Tari, F., Surfaces in ℝ4 and their projections to 3 spaces, Proc. Royal Soc. Edinb. 137A (2007), 1313–1328.Google Scholar
16.Rieger, J. H., Families of maps from the plane to the plane, J. Lond. Math. Soc. 36 (2) (1987), 351–369.Google Scholar
17.Rieger, J. H. and Ruas, M. A. S., Classification of $\mathcal{A}$-simple germs from k n to k 2, Compositio Math. 79 (1991), 99–108.Google Scholar
18.Tari, F., Projection of piecewise-smooth surfaces, J. Lond. Math. Soc. 44 (1) (1991), 155–172.Google Scholar
19.Wall, C. T. C., Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13 (1981), 481–539.Google Scholar