Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T13:52:52.908Z Has data issue: false hasContentIssue false
  • English
  • Français

The local model of an isotropic map-germ arising from one-dimensional symplectic reduction

Published online by Cambridge University Press:  24 October 2008

Goo Ishikawa
Goo Ishikawa
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we classify generic isotropic map-germs arising from the symplectic reduction process relative to a hypersurface (i.e. one-dimensional reduction), up to symplectic equivalence in the C category. These models are open Whitney umbrellas of arbitrary dimension and their suspensions. These singularities appear in the generalized Cauchy problem for HamiltonJacobi equations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 103 - 112
Copyright
Copyright © Cambridge Philosophical Society 1992

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.)

References

REFERENCES

1Abraham, R. and Marsden, J. E.. Foundation of Mechanics, 2nd edn. (Benjamin, 1978).Google Scholar
2Arnol'd, V. I.. Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities. Funktsional Anal. i Prilozhen. 6 (1972), 325.Google Scholar
3Arnol'd, V. I.. Singularities in variational calculus. J. Soviet Math. 27 (1984), 26792713.CrossRefGoogle Scholar
4Arnol'd, V. I.. Singularities of Caustics and Wave Fronts (Kluwer Academic Publishers, 1990).CrossRefGoogle Scholar
5Arnol'd, V. I., Gusein-Zade, S. M. and Varchenko, A. N.. Singularities of Differentiable Maps I (Birkhauser, 1985).CrossRefGoogle Scholar
6Boardman, M.. Singularities of differentiable maps. Inst. Hautes tudes Sci. Publ. Math. 33 (1967), 2157.CrossRefGoogle Scholar
7Brcker, Th.. Differentiable Germs and Catastrophes. London Math. Soc. Lecture Note Ser. no. 17 (Cambridge University Press, 1975).CrossRefGoogle Scholar
8Cleave, J.. The form of the tangent developable at points of zero torsion on space curves. Math. Proc. Cambridge Philos. Soc. 88 (1980), 403107.CrossRefGoogle Scholar
9Damon, J.. The Unfolding and Determinacy Theorems for Subgroups of ) and 0. Memoirs Amer. Math. Soc. no. 50 (American Mathematical Society, 1984).CrossRefGoogle Scholar
10Davydov, A. A.. The normal form of slow motions of an equation of relaxation type and fibrations of binormal surfaces. Math. USSR-Sb. 60 (1988), 133141.CrossRefGoogle Scholar
11Dufour, J. P.. Familles de courbes planes diffrentiables. Topology 22 (1983), 449474.CrossRefGoogle Scholar
12Fukuda, M. and Fukuda, T.. Algebra Q(f) determine the topological types of generic map germs. Invent. Math. 51 (1979), 231237.CrossRefGoogle Scholar
13Gibson, C. G.. Singular Points of Smooth Mappings (Pitman, 1979).Google Scholar
14Givental', A. B.. Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. Funktsional Anal, i Prilozhen. 20 (1986), 3541.Google Scholar
15Givental', A. B.. Singular Lagrangian varieties and their Lagrangian mappings. In Contemporary Problems of Mathematics, Itogi Nauki i Tekhniki, Ser. Sovrem. Probl. Mat. 33, (Vsesoyuz. Inst. Naukn. i Tekhn. Inform 1988), pp. 55112.Google Scholar
16Ishikawa, G.. Families of functions dominated by distributions of C-classes of mappings. Ann. Inst. Fourier (Grenoble) 33 (1983), 199217.CrossRefGoogle Scholar
17Ishikawa, G.. Parametrization of a singular Lagrangian variety. Trans. Amer. Math. Soc., to appear.Google Scholar
18Ishikawa, G.. Maslov class of an isotropic map-germ arising from one dimensional symplectic reduction. (Preprint.)Google Scholar
19Janeczko, S.. Generating families for images of Lagrangian submanifolds and open swallowtails. Math. Proc. Cambridge Philos. Soc. 100 (1986), 91107.CrossRefGoogle Scholar
20Malgrange, B.. Ideals of Differentiable Functions. (Oxford University Press, 1966).Google Scholar
21Malgrange, B.. Frobenius avec singularites, 2. Le cas general. Invent. Math. 39 (1977), 6789.CrossRefGoogle Scholar
22Mather, J. N.. Stability of C mappings IV: Classification of stable germs by ℝ algebras. Inst. Hautes tudes Sci. Publ. Math. 37 (1969), 223248.CrossRefGoogle Scholar
23Mather, J. N.. On ThomBoardman singularities. In Dynamical Systems (ed. Peixoto, M. M.) (Academic Press, 1973), pp. 233248.CrossRefGoogle Scholar
24Mather, J. N.. How to stratify mappings and jet spaces. In Singularits d'Applications Diffrentiables, Lecture Notes in Math. vol. 535 (Springer-Verlag, 1976), pp. 128176.CrossRefGoogle Scholar
25Mond, D.. Singularities of the tangent developable surface of a space curve. Quart. J. Math. Oxford Ser. (2) 40 (1989), 7991.CrossRefGoogle Scholar
26Morin, B.. Formes canonique des singularits d'une application diffrentiable, C.R. Acad. Sci. Paris 260 (1965), 56625665.Google Scholar
27Sherbak, O. P.. Projectively dual curves and Legendre singularities. Selecta Math. Soviet 5 (1986), 391421.Google Scholar
28Wall, C. T. C.. Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13 (1981), 481539.CrossRefGoogle Scholar
29Weinstein, A.. Lectures on Symplectic Manifolds. Regional Conf. Series in Math. no. 29 (American Mathematical Society, 1977).CrossRefGoogle Scholar