Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-13T13:38:58.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Singularities of smooth mappings with patterns*

Published online by Cambridge University Press:  26 March 2007

Kentaro Saji  and
Masatomo Takahashi
Kentaro Saji
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan (saji@math.sci.hokudai.ac.jp; takahashi@math.sci.hokudai.ac.jp)
Masatomo Takahashi
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan (saji@math.sci.hokudai.ac.jp; takahashi@math.sci.hokudai.ac.jp)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study smooth mappings with patterns which given by certain divergence diagrams of smooth mappings. The divergent diagrams of smooth mappings can be regard as smooth mappings from manifolds with singular foliations. Our concerns are generic differential topology and generic smooth mappings with patterns. We give a generic semi-local classification of surfaces with singularities and patterns as an application of singularity theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 2 , 2007 , pp. 415 - 430
Copyright
Copyright © Royal Society of Edinburgh 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Dedicated to Professor Takao Matumoto on the occasion of his sixtieth birthday.

References

1 Arnol'd, V. I.. Wave front evolution and equivalent Morse lemma. Commun. Pure Appl. Math. 29 (1976), 557–582.Google Scholar
2 Bleecker, D. and Wilson, L.. Stability of Gauss maps. Illinois J. Math. 22 (1978), 279–289.Google Scholar
3 Boardman, J.. Singularities of differentiable maps. Publ. Math. IHES 33 (1967), 21–57.Google Scholar
4 Bruce, J. W.. Seeing: the mathematical viewpoint. Math. Intelligencer 6 (1984), 18–25.Google Scholar
5 Dufour, J. P.. Families de courbes planes differentiables. Topology 22 (1983), 449–474.Google Scholar
6 Dufour, J. P.. Modules pour les families de courbes planes. Annls Inst. Fourier 39 (1989), 225–238.Google Scholar
7 Dufour, J. P. and Jean, P.. Rigidity of webs and families of hypersurfaces. In Singularities and dynamical systems (ed. Pnevmatikos, S. N.), pp. 271–283 (Elsevier, 1985).Google Scholar
8 Dufour, J. P. and Jean, P.. Families de surfaces differentiables. J. bond. Math. Soc. (2) 42 (1990), 175–192.Google Scholar
9 Dufour, J. P. and Kurokawa, Y.. Topological rigidity for certain families of differentiable plane curves. J. Lond. Math. Soc. 65 (2002), 223–242.Google Scholar
10 Dufour, J. P. and Tueno, J. P.. Singularités et photographies de surfaces. J. Geom. Phys. 9 (1992), 173–182.Google Scholar
11 Gibson, C. G.. Singular points of smooth mappings (Pitman, 1979).Google Scholar
12 Golubitsky, M. and Guillemin, V.. Stable mappings and their singularities. Graduate Texts in Mathematics, vol. 14 (Springer, 1973).Google Scholar
13 Izumiya, S. and Kurokawa, Y.. Holonomic systems of Clairaut type. Diff. Geom. Applic. 5 (1995), 219–235.Google Scholar
14 Izumiya, S. and Marar, W. L.. On topologically stable singular surfaces in a 3-manifold. J. Geom. 52 (1995), 108–119.Google Scholar
15 Izumiya, S. and Maruyama, K.. Transversal Whitney topology and singularities of Haefliger foliations. In Real and complex singularities, Lecture Notes in Pure and Applied Mathematics, vol. 232, pp. 165–173 (Marcel Dekker, 2003).Google Scholar
16 Izumiya, S. and Takeuchi, N.. Singularities of ruled surfaces in ℝ3 . Math. Proc. Camb. Phil. Soc. 130 (2001), 1–11.Google Scholar
17 Kurokawa, Y.. Singularities for projections of contour lines of surfaces onto planes. Hokkaido Math. J. 30 (2001), 573–587.Google Scholar
18 Mancini, S., Ruas, M. A. S. and Teixeira, M. A.. On divergent diagrams of finite codimension. Portugaliae Math. 59 (2002), 179–194.Google Scholar
19 Mather, J.. Stability of C ∞-mappings. V. Transversality. Adv. Math. 4 (1970), 301–336.Google Scholar
20 Nishimura, T.. A constructive method to get right–left equivalence for smooth map germs and its application to divergent diagrams. Mat. Contemp. 5 (1993), 137–160.Google Scholar
21 Saji, K.. Singularities of non-degenerate n-ruled (n + 1)-manifolds in Euclidean space. Banach Center Publ. 65 (2004), 211–225.Google Scholar
22 Saji, K. and Takahashi, M.. Topological properties of smooth mapping with patterns. (In preparation.)Google Scholar
23 Takahashi, M.. Holonomic systems of general Clairaut type. Hokkaido Math. J. 34 (2005), 247–263.Google Scholar
24 Wall, C. T. C.. Openness and rnultitransversality. In Real and complex singularities, Lecture Notes in Pure and Applied Mathematics, vol. 232, pp. 121–136 (Marcel Dekker, 2003).Google Scholar
25 Whitney, H.. The singularities of smooth n-manifold in (2n – 1)-space. Ann. Math. 45 (1944), 247–293.Google Scholar
26 Whitney, H.. Elementary structure of real algebraic varieties. Ann. Math. 66 (1957), 545–556.Google Scholar