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Parametrization of a singular Lagrangian variety


Author: Goo Ishikawa
Journal: Trans. Amer. Math. Soc. 331 (1992), 787-798
MSC: Primary 58C27
DOI: https://doi.org/10.1090/S0002-9947-1992-1044961-9
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.


References [Enhancements On Off] (What's this?)

  • [1] R. Abraham and J. E. Marsden, Foundation of mechanics, 2nd ed., Benjamin, New York, 1978.
  • [2] S. B. Alexander, I. D. Berg, and R. L. Bishop, Cauchy uniqueness in the Riemannian obstacle problem, Lecture Notes in Math., vol. 1209, Springer-Verlag, 1986, pp. 1-7. MR 863742 (88e:53064)
  • [3] V. I. Arnol'd, Lagrangian manifolds with singularities, asymptotic rays and the open swallowtail, Funct. Anal. Appl. 15 (1981), 235-246. 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, Differential germs and catastrophes, London Math. Soc. Lecture Notes Series 17, Cambridge Univ. Press, 1975. MR 0494220 (58:13132)
  • [6] A. B. Givental', Manifolds of polynomials having a root of fixed multiplicity, and the generalized Newton equation, Funct. Anal. Appl. 16 (1982), 10-14. 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. W. Guillemin, Stable mappings and their singularities, Springer-Verlag, 1973. MR 0341518 (49:6269)
  • [9] G. Ishikawa, Families of functions dominated by distributions of $ \mathcal{C}$-classes of mappings, Ann. Inst. Fourier (Grenoble) 33 (1983), 199-217. MR 699495 (84g:58014)
  • [10] S. Janeczko, Generating families for images of Lagrangian submanifolds and open swallowtails, Math. Proc. Cambridge Philos. Soc. 100 (1986), 91-107. MR 838655 (87h:58022)
  • [11] -, Constrained Lagrangian submanifolds over singular constraining varieties and discriminant varieties, Ann. Inst. Henri Poincaré46 (1987), 1-26. MR 877993 (88d:58036)
  • [12] -, Tensor invariants and invariant symplectic geometry of binary forms, Bull. Polon. Acad. Sci. Math. 36 (1988), 15-23. MR 998203 (91e:05014)
  • [13] J. Keller, A geometrical theory of diffraction, Proc. Sympos. Appl. Math., vol. 8, Amer. Math. Soc., Providence, R.I., 1958, pp. 27-52. MR 0094120 (20:640)
  • [14] B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press, 1966. MR 0212575 (35:3446)
  • [15] -, Frobenius avec singularités, $ 2$. Le cas général, Invent. Math. 39 (1977), 67-89. MR 0508170 (58:22685b)
  • [16] J. N. Mather, Notes on topological stability, Harvard Univ. preprint, 1970.
  • [17] A. N. Varchenko and A. B. Givental', Period mapping and intersection form, Funktsional Anal, i Prilozhen 16 (1982), 7-20=Functional Anal. Appl. 16 (1982), 83-93. MR 659161 (84b:32016)
  • [18] A. Weinstein, Lectures on symplectic manifolds, CBMS Regional Conf. Ser. in Math., no. 29, Amer. Math. Soc., Providence, R.I., 1977. MR 0464312 (57:4244)
  • [19] H. Whitney, Tangents to an analytic variety, Ann. of Math. (2) 81 (1964), 496-549. MR 0192520 (33:745)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1044961-9
Keywords: Lagrangian variety, isotropic map, parametrization, stratification
Article copyright: © Copyright 1992 American Mathematical Society

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