PROJECTIONS OF HYPERSURFACES IN R4 WITH BOUNDARY TO PLANES

MARTINS, LUCIANA F. ; NABARRO, ANA CLAUDIA

Glasgow mathematical journal, Tome 56 (2014) no. 1, pp. 149-167

Résumé

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.

Détail

DOI : 10.1017/S001708951300013X

Mots-clés : 53A07, 58K05, 58K40

Bibliographie

[1] 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). Google Scholar | DOI

[2] 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] 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] 4.Bruce, J. W. and Giblin, P. J., Projections of surfaces with boundary, Proc. Lond. Math. Soc. 60 (3) (1990), 392–416. Google Scholar | DOI

[5] 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. Google Scholar | DOI

[6] 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] 7.Damon, J. N., The unfolding and determinacy theorems for subgroups of and , Mem. Amer. Math. Soc. 50 (306) (1984), 1–88. Google Scholar

[8] 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. Google Scholar | DOI

[9] 9.Kirk, N. P., Computational aspects of classifying singularities, LMS J. Comput. Math. 3 (2000), 207–228. Google Scholar | DOI

[10] 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. Google Scholar | DOI

[11] 11.Mather, J. N., Stability of C∞ mappings, IV: Classification of stable germs by -algebras, Inst. Hautes Etudes Sci. Publ. Math. 37 (1969), 223–248. Google Scholar | DOI

[12] 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. Google Scholar | DOI

[13] 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] 14.Nabarro, A. C., Duality and contact of hypersurfaces in ℝ4 with hyperplanes and lines, Proc. Edinb. Math Soc. 46 (2003), 637–648. Google Scholar | DOI

[15] 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] 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] 17.Rieger, J. H. and Ruas, M. A. S., Classification of -simple germs from k n to k 2, Compositio Math. 79 (1991), 99–108. Google Scholar

[18] 18.Tari, F., Projection of piecewise-smooth surfaces, J. Lond. Math. Soc. 44 (1) (1991), 155–172. Google Scholar

[19] 19.Wall, C. T. C., Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13 (1981), 481–539. Google Scholar