Global stability for diagrams of differentiable applications

Luis Antonio Favaro; C. M. Mendes

Annales de l'institut Fourier (1986)

  • Volume: 36, Issue: 1, page 133-153
  • ISSN: 0373-0956

Abstract

top
In this paper, we give some examples which point to the non-existence of C ∞ -global stable diagrams R ← g M → f R , M compact. If Φ : M → Q is fixed we define the Φ -equivalence for maps f : M → P and the corresponding Φ -stability. The globalization procedure works and we can compare the Φ -stability, Φ -infinitesimal stability, and Φ -homotopical stability. Also we give some characterization theorems for lower dimensions.

How to cite

top

Favaro, Luis Antonio, and Mendes, C. M.. "Global stability for diagrams of differentiable applications." Annales de l'institut Fourier 36.1 (1986): 133-153. <http://eudml.org/doc/74701>.

@article{Favaro1986,
abstract = {In this paper, we give some examples which point to the non-existence of $C^\infty $-global stable diagrams $R\mathrel \{\mathop \{\hspace\{0.0pt\}\leftarrow \}\limits ^\{g\}\}M \mathrel \{\mathop \{\hspace\{0.0pt\}\rightarrow \}\limits ^\{f\}\}R$, $M$ compact. If $\Phi $ : $M\rightarrow Q$ is fixed we define the $\Phi $-equivalence for maps $f: M\rightarrow P$ and the corresponding $\Phi $-stability. The globalization procedure works and we can compare the $\Phi $-stability, $\Phi $-infinitesimal stability, and $\Phi $-homotopical stability. Also we give some characterization theorems for lower dimensions.},
author = {Favaro, Luis Antonio, Mendes, C. M.},
journal = {Annales de l'institut Fourier},
keywords = {stability of diagrams; unfoldings; singularities of differentiable mappings},
language = {eng},
number = {1},
pages = {133-153},
publisher = {Association des Annales de l'Institut Fourier},
title = {Global stability for diagrams of differentiable applications},
url = {http://eudml.org/doc/74701},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Favaro, Luis Antonio
AU - Mendes, C. M.
TI - Global stability for diagrams of differentiable applications
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 1
SP - 133
EP - 153
AB - In this paper, we give some examples which point to the non-existence of $C^\infty $-global stable diagrams $R\mathrel {\mathop {\hspace{0.0pt}\leftarrow }\limits ^{g}}M \mathrel {\mathop {\hspace{0.0pt}\rightarrow }\limits ^{f}}R$, $M$ compact. If $\Phi $ : $M\rightarrow Q$ is fixed we define the $\Phi $-equivalence for maps $f: M\rightarrow P$ and the corresponding $\Phi $-stability. The globalization procedure works and we can compare the $\Phi $-stability, $\Phi $-infinitesimal stability, and $\Phi $-homotopical stability. Also we give some characterization theorems for lower dimensions.
LA - eng
KW - stability of diagrams; unfoldings; singularities of differentiable mappings
UR - http://eudml.org/doc/74701
ER -

References

top
  1. [1] M.A. BUCHNER, Stability of the cut locus in dimensions less than or equal to 6, Inventiones Math., 43 (1977), 199-231. Zbl0365.58010MR58 #2866
  2. [2] J.W. BRUCE, On singularities, envelopes and elementary differential geometry, Math. Proc. Camb. Phil. Soc., (1981). Zbl0454.58002MR82b:58018
  3. [3] M.J.D. CARNEIRO, On the Envelope Theory, PhD Thesis, Princeton, (1980). 
  4. [4] J.P. DUFOUR, Déploiements de cascades d'applications différentiables, C.R.A.S., Paris, 281 (1975), A 31-34. Zbl0317.58005MR52 #6775
  5. [5] J.P. DUFOUR, Diagrammes d'applications différentiables, Thèse Université des Sciences et Techniques du Languedoc, (1979). Zbl0354.58011
  6. [6] J.P. DUFOUR, Stabilité simultanée de deux fonctions, Ann. Inst. Fourier, Grenoble, 29, 1 (1979), 263-282. Zbl0364.58007MR80f:58010
  7. [7] L.A. FAVARO and C.M. MENDES, Singularidades e Envoltorias, Comunicaçao, IV Escola de Geometria Diferencial, IMPA, Rio de Janeiro, (1982). 
  8. [8] M. GOLUBITSKY and V. GUILLEMIN, Stable Mappings and Their Singularities, Graduate Texts in Mathematics, Springer-Verlag, Vol. 14 (1973). Zbl0294.58004MR49 #6269
  9. [9] J.N. MATHER, Stability of C∞ mappings II : Infinitesimal, stability implies stability, Annals of Math., Vol. 89, n° 2 (1969). Zbl0177.26002MR41 #4582
  10. [10] J. MARTINET, Déploiements versels des applications différentiables et classification des applications stables, Lectures Notes in Mathematics, 535 (1975). Zbl0362.58004
  11. [11] C.M. MENDES, ψ-Estabilidade, Tese de Doutorado, ICMSC-USP, (1981). 
  12. [12] R. THOM, Sur la théorie des enveloppes, J. Math. Pure et Appl., Tome XLI, Fac. 2 (1962). Zbl0105.16102MR25 #4454

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.