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  • Cited by 11
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Izumiya, S. and Janeczko, S. 2003. A symplectic framework for multiplane gravitational lensing. Journal of Mathematical Physics, Vol. 44, Issue. 5, p. 2077.


    Bogaevski, I.A. and Ishikawa, G. 2002. Lagrange mappings of the first open Whitney umbrella. Pacific Journal of Mathematics, Vol. 203, Issue. 1, p. 115.


    Janeczko, S. 2000. Lagrangian submanifolds in product symplectic spaces. Journal of Mathematical Physics, Vol. 41, Issue. 8, p. 5642.


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    Janeczko, S. 1992. Covariant symplectic geometry of binary forms and singularities of systems of rays. Reports on Mathematical Physics, Vol. 31, Issue. 2, p. 147.


    Janeczko, S. and Stewart, Ian 1991. Singularity Theory and its Applications.


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    Janeczko, S. 1990. Generalized Luneburg canonical varieties and vector fields on quasicaustics. Journal of Mathematical Physics, Vol. 31, Issue. 4, p. 997.


    Janeczko, Stanisław and Kowalczyk, Adam 1988. Isotropic varieties in the singular symplectic geometry. Bulletin of the Australian Mathematical Society, Vol. 38, Issue. 02, p. 161.


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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 1
  • July 1986, pp. 91-107

Generating families for images of Lagrangian submanifolds and open swallowtails

  • Stanisław Janeczko (a1)
  • DOI: http://dx.doi.org/10.1017/S0305004100065889
  • Published online: 24 October 2008
Abstract
Summary

In this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL2(ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.

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V. I. Arnold . Lagrangian manifolds with singularities, asymptotic rays and the open swallowtail. Functional Anal. Appl. 15 (1981), 235246.

[7]J. Duistermaat . Oscillatory integrals, lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27 (1974), 207281.

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[16]J. Sniatycki and W. M. Tulczyjew . Generating forms of lagrangian submanifolds. Indiana Univ. Math. J. 22 (1972), 267275.

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[22]V. M. Zakalyukin . On lagrangian and legendrian singularities. Functional Anal. Appl. 10 (1976), 2331.

[23]E. C. Zeeman and D. J. A. Trotman . The classification of elementary catastrophes of codimension ≤ 5. In Structural Stability, the Theory of Catastrophes and Applications in the Sciences. Lecture Notes in Math. vol. 525 (Springer-Verlag1976), 263327.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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