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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Parametrization of a singular Lagrangian variety


Author: Goo Ishikawa
Journal: Trans. Amer. Math. Soc. 331 (1992), 787-798
MSC: Primary 58C27
MathSciNet review: 1044961
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Abstract: We give stabilization and parametrization theorems for a class of singular varieties in the space of polynomials of one variable and generalize the results of Arnol'd and Givental'. The class contains the open swallowtails and the open Whitney umbrella. The parametrization is associated with the singularity of a stable mapping (in the sense of Thom and Mather) of kernel rank one.


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  • [1] R. Abraham and J. E. Marsden, Foundation of mechanics, 2nd ed., Benjamin, New York, 1978.
  • [2] Stephanie B. Alexander, I. David Berg, and Richard L. Bishop, Cauchy uniqueness in the Riemannian obstacle problem, Differential geometry, Peñíscola 1985, Lecture Notes in Math., vol. 1209, Springer, Berlin, 1986, pp. 1–7. MR 863742 (88e:53064), http://dx.doi.org/10.1007/BFb0076617
  • [3] V. I. Arnol′d, Lagrangian manifolds with singularities, asymptotic rays and the unfurled swallowtail, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 1–14, 96 (Russian). MR 639196 (83c:58011)
  • [4] -, Singularities in variational calculus, Itogi Nauki, Contemporary Problems in Mathematics 22 (1983), 3-55; English transl., J. Soviet Math. 27 (1984), 2679-2713.
  • [5] Th. Bröcker, Differentiable germs and catastrophes, Cambridge University Press, Cambridge-New York-Melbourne, 1975. Translated from the German, last chapter and bibliography by L. Lander; London Mathematical Society Lecture Note Series, No. 17. MR 0494220 (58 #13132)
  • [6] A. B. Givental′, Manifolds of polynomials that have a root of fixed comultiplicity, and the generalized Newton equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 1, 13–18, 96 (Russian). MR 648804 (83j:58020)
  • [7] -, Lagrangian imbeddings of surfaces and unfolded Whitney umbrella, Funct. Anal. Appl. 20 (1986), 197-203.
  • [8] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. MR 0341518 (49 #6269)
  • [9] Goo Ishikawa, Families of functions dominated by distributions of \cal𝐶-classes of mappings, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 199–217. MR 699495 (84g:58014)
  • [10] Stanisław Janeczko, Generating families for images of Lagrangian submanifolds and open swallowtails, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 91–107. MR 838655 (87h:58022), http://dx.doi.org/10.1017/S0305004100065889
  • [11] Stanisław Janeczko, Constrained Lagrangian submanifolds over singular constraining varieties and discriminant varieties, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 1–26 (English, with French summary). MR 877993 (88d:58036)
  • [12] Stanisław Janeczko, Tensor invariants and invariant symplectic geometry of binary forms, Bull. Polish Acad. Sci. Math. 36 (1988), no. 1-2, 15–23 (English, with Russian summary). MR 998203 (91e:05014)
  • [13] Joseph B. Keller, A geometrical theory of diffraction, Calculus of variations and its applications. Proceedings of Symposia in Applied Mathematics, Vol. 8, For the American Mathematical Society: McGraw-Hill Book Co., Inc., New York-Toronto-London, 1958, pp. 27–52. MR 0094120 (20 #640)
  • [14] B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575 (35 #3446)
  • [15] B. Malgrange, Frobenius avec singularités. II. Le cas général, Invent. Math. 39 (1977), no. 1, 67–89 (French). MR 0508170 (58 #22685b)
  • [16] J. N. Mather, Notes on topological stability, Harvard Univ. preprint, 1970.
  • [17] A. N. Varchenko and A. B. Givental′, The period mapping and the intersection form, Funktsional. Anal. i Prilozhen. 16 (1982), no. 2, 7–20, 96 (Russian). MR 659161 (84b:32016)
  • [18] Alan Weinstein, Lectures on symplectic manifolds, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976; Regional Conference Series in Mathematics, No. 29. MR 0464312 (57 #4244)
  • [19] Hassler Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496–549. MR 0192520 (33 #745)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1044961-9
PII: S 0002-9947(1992)1044961-9
Keywords: Lagrangian variety, isotropic map, parametrization, stratification
Article copyright: © Copyright 1992 American Mathematical Society