Umbilical characteristic number of Lagrangian mappings of 3-dimensional pseudooptical manifolds

Maxim È. Kazarian

Banach Center Publications (1996)

  • Volume: 33, Issue: 1, page 161-170
  • ISSN: 0137-6934

Abstract

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As shown by V. Vassilyev [V], D 4 ± singularities of arbitrary Lagrangian mappings of three-folds form no integral characteristic class. We show, nevertheless, that in the pseudooptical case the number of D 4 ± singularities counted with proper signs forms an invariant. We give a topological interpretation of this invariant, and its applications. The results of the paper may be considered as a 3-dimensional generalization of the results due to V. I. Arnold [A].

How to cite

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Kazarian, Maxim È.. "Umbilical characteristic number of Lagrangian mappings of 3-dimensional pseudooptical manifolds." Banach Center Publications 33.1 (1996): 161-170. <http://eudml.org/doc/262730>.

@article{Kazarian1996,
abstract = {As shown by V. Vassilyev [V], $D_4^±$ singularities of arbitrary Lagrangian mappings of three-folds form no integral characteristic class. We show, nevertheless, that in the pseudooptical case the number of $D_4^±$ singularities counted with proper signs forms an invariant. We give a topological interpretation of this invariant, and its applications. The results of the paper may be considered as a 3-dimensional generalization of the results due to V. I. Arnold [A].},
author = {Kazarian, Maxim È.},
journal = {Banach Center Publications},
keywords = {singularities; Lagrangian mappings; invariant},
language = {eng},
number = {1},
pages = {161-170},
title = {Umbilical characteristic number of Lagrangian mappings of 3-dimensional pseudooptical manifolds},
url = {http://eudml.org/doc/262730},
volume = {33},
year = {1996},
}

TY - JOUR
AU - Kazarian, Maxim È.
TI - Umbilical characteristic number of Lagrangian mappings of 3-dimensional pseudooptical manifolds
JO - Banach Center Publications
PY - 1996
VL - 33
IS - 1
SP - 161
EP - 170
AB - As shown by V. Vassilyev [V], $D_4^±$ singularities of arbitrary Lagrangian mappings of three-folds form no integral characteristic class. We show, nevertheless, that in the pseudooptical case the number of $D_4^±$ singularities counted with proper signs forms an invariant. We give a topological interpretation of this invariant, and its applications. The results of the paper may be considered as a 3-dimensional generalization of the results due to V. I. Arnold [A].
LA - eng
KW - singularities; Lagrangian mappings; invariant
UR - http://eudml.org/doc/262730
ER -

References

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  1. [A] V. I. Arnold, Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques, Cahiers Math. Décision, CEREMADE, 9320, 14/6/93, 9 pp. 
  2. [AG] V. I. Arnold and A. V. Givental, Symplectic Geometry, in: Encyclopedia of Math. Sci. Dynamical Systems 4, Springer, 1990, 4-136. 
  3. [Ch] Yu. V. Chekanov, Caustics in geometrical optics, Funct. Anal. Appl. 20 (1986), 223-226. Zbl0622.58004
  4. [D] G. Darboux, Leçons sur la théorie générale des surfaces, Vol. 4, Gauthier-Villars, Paris, 1896. Zbl27.0497.01
  5. [F] D. B. Fuchs, Maslov-Arnol'd characteristic classes, Soviet Math. Dokl. 9 (1968), 96-99. Zbl0175.20304
  6. [P1] I. R. Porteous, The normal singularities of surfaces in ℝ 3 , in: Proc. Sympos. Pure Math. 40, Part 2, Amer. Math. Soc., 1983, 379-393. 
  7. [P2] I. R. Porteous, Geometric Differentiation, Cambridge University Press, 1994. 
  8. [V] V. A. Vassilyev, Lagrange and Legendre Characteristic Classes, Gordon and Breach, 1988. 
  9. [Z] V. M. Zakalyukin, Rearrangements of fronts and caustics depending on a parameter, and versality of maps, in: Current Problems of Math. 22, VINITI, 1983, 56-93. 

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