Journal of Soviet Mathematics

, Volume 52, Issue 4, pp 3246–3278 | Cite as

Singular Lagrangian manifolds and their Lagrangian maps

  • A. B. Givental'
Article

Abstract

The paper examines the singularity theory of Lagrangian manifolds and its connection with variational calculus, classification of Coxeter groups, and symplectic topology. We consider the application of the theory to the problem of going past an obstacle, to partial differential equations, and to the analysis of singularities of ray systems.

Keywords

Differential Equation Manifold Partial Differential Equation Singularity Theory Coxeter Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    V. I. Arnol'd, Mathematical Principles of Classical Mechanics [in Russian], Nauka, Moscow (1974).Google Scholar
  2. 2.
    V. I. Arnol'd, “Normal forms of functions near degenerate critical points, Weyl groups Ak, Dk, Ek, and Lagrangian singularities,” Funkts. Anal. Prilozhen.,6, No. 4, 3–25 (1972).Google Scholar
  3. 3.
    V. I. Arnol'd, “Critical points of functions on a manifold with an edge, simple Lie groups Bk, Ck, F4, and singularities of evolutes,” Usp. Mat. Nauk,33, No. 5, 91–105 (1978).Google Scholar
  4. 4.
    V. I. Arnol'd, “Lagrangian manifolds with singularities, asymptotic rays, and the open swallowtail,” Funkts. Anal. Prilozhen.,15, No. 4, 1–14 (1981).Google Scholar
  5. 5.
    V. I. Arnol'd, “Potential flows of collisionfree particles, Lagrangian singularities, and metamorphoses of caustics,” Tr. Seminara im. I. G. Petrovskogo,8, No. 5–53 (1982).Google Scholar
  6. 6.
    V. I. Arnol'd, “Singularities of ray systems,” Usp. Mat. Nauk,38, No. 2, 77–147 (1983).Google Scholar
  7. 7.
    V. I. Arnol'd, “Singularities in variational calculus,” Itogi Nauki i Tekhniki, Sovr. Probl. Mat.,22, 3–55 (1983).Google Scholar
  8. 8.
    V. I. Arnol'd, “Singularities in variational calculus,” Usp. Mat. Nauk,39, No. 5, 256 (1984).Google Scholar
  9. 9.
    V. I. Arnol'd, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps [in Russian], Vol. 1, Nauka, Moscow (1982).Google Scholar
  10. 10.
    A. I. Arnol'd and A. B. Givental', “Symplectic geometry,” Itogi Nauki i Tekhniki, Sovr. Probl. Mat., Fundam. Napravl., Vol. 4, 5–139 (1985).Google Scholar
  11. 11.
    E. Brieskorn, “Monodromy of isolated singularities of hypersurfaces,” Matematika [Russian translations],15, No. 4, 13–161 (1971).Google Scholar
  12. 12.
    E. Brieskorn, “On braid groups (after V. I. Arnol'd),” Matematika [Russian translations],18, No. 3, 49–59 (1974).Google Scholar
  13. 13.
    N. Bourbaki, Groups and Lie Algebras [Russian translation], Mir, Moscow (1978), Chaps. 4–6.Google Scholar
  14. 14.
    A. N. Varchenko, “Asymptotic Hodge structure in vanishing cohomologies,” Izv. Akad. Nauk SSSR, Ser. Mat.,45, No. 3, 540–591 (1981).Google Scholar
  15. 15.
    A. N. Varchenko, “On local residue and the intersection form in vanishing cohomologies,” Izv. Akad. Nauk SSSR, Ser., Mat.,48, No. 5, 1010–1035 (1984).Google Scholar
  16. 16.
    A. N. Varchenko and A. B. Givental', “Period mapping and intersection form,” Funkts. Anal. Prilozhen.,16, No. 2, 7–20 (1982).Google Scholar
  17. 17.
    A. G. Gasparyan, “Application of multidimensional matrices to the analysis of polynomials,” Dokl. Akad. Nauk ArmSSR,70, No. 3, 133–141 (1980).Google Scholar
  18. 18.
    A. B. Givental', “Manifolds of polynomials having a root of fixed comultiplicity and generalized Newton's equation,” Funkts. Anal. Prilozhen.,16, No. 2, 13–18 (1982).Google Scholar
  19. 19.
    A. B. Givental', “Lagrangian manifolds with singularities and irreducible SL2-modules,” Usp. Mat. Nauk,38, No. 6, 109–110 (1983).Google Scholar
  20. 20.
    A. B. Givental', “Lagrangian embeddings of surfaces and the open Whitney umbrella,” Funkts. Anal. Prilozhen.,20, No. 3, 35–41 (1986).Google Scholar
  21. 21.
    V. V. Goryunov, “Singularities of projections of complete intersections,” Itogi Nauki i Tekhniki, Sovr. Probl. Mat.,22, 167–206 (1983).Google Scholar
  22. 22.
    V. M. Zakalyukin, “Evolution of fronts and caustics dependent on a parameter and versality of maps,” Itogi Nauki i Tekhniki, Sovr. Probl. Mat.,22, 53–96 (1983).Google Scholar
  23. 23.
    V. M. Zakalyukin, “A generalization of Lagrangian triads,” Usp. Mat. Nauk,41, No. 4, 180 (1986).Google Scholar
  24. 24.
    V. P. Kostov, “Versal deformations of differential forms of degreeα on the straight line,” Usp. Mat. Nauk,40, No. 5, 235 (1985).Google Scholar
  25. 25.
    S. K. Lando, “Normal forms of powers of volume forms,” Funkts. Anal. Prilozhen.,19, No. 2, 78–79 (1985).Google Scholar
  26. 26.
    V. V. Lychagin, “Geometric singularities of solutions of nonlinear differential equations,” Dokl. Akad. Nauk SSSR,261, No. 6, 1299–1303 (1981).Google Scholar
  27. 27.
    O. V. Lyashko, “The geometry of bifurcation diagrams,” Itogi Nauki i Tekhniki, Sovr. Probl. Mat.,22, 94–129 (1983).Google Scholar
  28. 28.
    V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Moscow State Univ. (1965).Google Scholar
  29. 29.
    I. G. Shcherbak, “Duality of edge singularities,” Usp. Mat. Nauk,39, No. 2, 207–208 (1984).Google Scholar
  30. 30.
    O. P. Shcherbak, “Projectively dual spatial curves and Legendrian singularities,” Trudy Tbilis. Univ.,232–233, 13–14Google Scholar
  31. 31.
    O. P. Shcherbak, “Singularities of the family of evolvents in the neighborhood of the inflection point of a curve and the group H3 generated by reflections,” Funkts. Anal. Prilozhen.,17, No. 4, 70–72 (1983).Google Scholar
  32. 32.
    O. P. Shcherbak, “Wave fronts and reflection groups,” Usp. Mat. Nauk,43, No. 3, 125–160 (1988).Google Scholar
  33. 33.
    D. Bennequin, Caustique mystique, Seminaire N. Bourbaki, No. 634 (1984).Google Scholar
  34. 34.
    J. W. Bruce, “Vector fields on discriminants,” Bull. Lond. Math. Soc.,7, No. 3, 257–262 (1985).Google Scholar
  35. 35.
    M. Gromov, “Pseudo-holomorphic curves in symplectic manifolds,” Invent. Math.,82, 307–347 (1985).Google Scholar
  36. 36.
    E. Looijenga, “A period mapping for certain semi-universal deformations,” Compos. Math.,30, 299–316 (1974).Google Scholar
  37. 37.
    J. Milnor and P. Orlic, “Isolated singularities defined by weighted homogeneous polynomials,” Topology,9, No. 2, 385–393 (1970).Google Scholar
  38. 38.
    B. Morin, “Formes canonique des singularites d'une application differentiable,” C. R. Acad. Sci. Paris,260, 5663–5665, 6503–6506 (1965).Google Scholar
  39. 39.
    J. Moser, “On the volume elements on manifolds,” Trans. AMS,120, No. 2, 280–296 (1965).Google Scholar
  40. 40.
    Nguenhuu Duc and Nguyen tien Dai, “Stabilite de l'interaction geometrique entre deux composantes,” C. R. Acad. Sci. Paris,291, 113–116 (1980).Google Scholar
  41. 41.
    K. Saito, “Quasihomogene isolierte Singularitaten von Hyperflachen,” Invent. Math.,14, No. 2, 123–142 (1971).Google Scholar
  42. 42.
    K. Saito, “On a linear structure of a quotient variety by a finite reflection group,” Preprint, Kyoto Univ. (1979).Google Scholar
  43. 43.
    P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lect. Notes. Math.,815 (1980).Google Scholar
  44. 44.
    A. Weinstein, Lectures of Symplectic Manifolds, Providence, RI (1977).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • A. B. Givental'

There are no affiliations available

Personalised recommendations