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

• Autorzy: Maxim È. Kazarian
• Języki publikacji: EN

Abstrakty

EN

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].

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1996
• Tom: 33
• Numer: 1
• Strony: 161-170

Daty

wydano

1996

Twórcy

autor

Maxim È. Kazarian

• Moscow Independent University, Altayskaya St., 34-145, Moscow 107 589, Russia

Bibliografia

• [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.
• [D] G. Darboux, Leçons sur la théorie générale des surfaces, Vol. 4, Gauthier-Villars, Paris, 1896.
• [F] D. B. Fuchs, Maslov-Arnol'd characteristic classes, Soviet Math. Dokl. 9 (1968), 96-99.
• [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.
• [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.