Skip to main content
×
Home
    • Aa
    • Aa

Generating families for images of Lagrangian submanifolds and open swallowtails

  • Stanisław Janeczko (a1)
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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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.

[8] M. Golubitsky and V. W. Guillemin . Stable mappings and their singularities (Springer-Verlag, 1973).

[9] M. Golubitsky and V. W. Guillemin . Contact equivalence for lagrangian manifolds. Adv. in Math. 15 (1975), 375387.

[15] R. B. Melrose . Equivalence of glancing hypersurfaces. Invent. Math. 37 (1976), 155191.

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

[20] A. Weinstein . Symplectic manifolds and their lagrangian submanifolds. Adv. in Math. 6 (1971), 329349.

[21] G. Wassermann . Stability of unfoldings in space and time. Acta Math. 135 (1975), 58128

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 29 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th May 2017. This data will be updated every 24 hours.