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

Banach Center Publications (1996)

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

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topKazarian, 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

top- [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.
- [AG] V. I. Arnold and A. V. Givental, Symplectic Geometry, in: Encyclopedia of Math. Sci. Dynamical Systems 4, Springer, 1990, 4-136.
- [Ch] Yu. V. Chekanov, Caustics in geometrical optics, Funct. Anal. Appl. 20 (1986), 223-226. Zbl0622.58004
- [D] G. Darboux, Leçons sur la théorie générale des surfaces, Vol. 4, Gauthier-Villars, Paris, 1896. Zbl27.0497.01
- [F] D. B. Fuchs, Maslov-Arnol'd characteristic classes, Soviet Math. Dokl. 9 (1968), 96-99. Zbl0175.20304
- [P1] I. R. Porteous, The normal singularities of surfaces in ${\mathbb{R}}^{3}$, in: Proc. Sympos. Pure Math. 40, Part 2, Amer. Math. Soc., 1983, 379-393.
- [P2] I. R. Porteous, Geometric Differentiation, Cambridge University Press, 1994.
- [V] V. A. Vassilyev, Lagrange and Legendre Characteristic Classes, Gordon and Breach, 1988.
- [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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