Singularity Classes of Special 2-Flags

Piotr Mormul

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

In the paper we discuss certain classes of vector distributions in the tangent bundles to manifolds, obtained by series of applications of the so-called generalized Cartan prolongations (gCp). The classical Cartan prolongations deal with rank-2 distributions and are responsible for the appearance of the Goursat distributions. Similarly, the so-called special multi-flags are generated in the result of successive applications of gCp's. Singularities of such distributions turn out to be very rich, although without functional moduli of the local classification. The paper focuses on special 2-flags, obtained by sequences of gCp's applied to rank-3 distributions. A stratification of germs of special 2-flags of all lengths into singularity classes is constructed. This stratification provides invariant geometric significance to the vast family of local polynomial pseudo-normal forms for special 2-flags introduced earlier in [Mormul P., Banach Center Publ., Vol. 65, Polish Acad. Sci., Warsaw, 2004, 157–178]. This is the main contribution of the present paper. The singularity classes endow those multi-parameter normal forms, which were obtained just as a by-product of sequences of gCp's, with a geometrical meaning.

Détail

Keywords: generalized Cartan prolongation; special multi-flag; singularity class.

Bibliographie

[1] Adachi J., “Global stability of special multi-flags”, Israel J. Math. (to appear)

[2] Agrachev A. A., “Feedback-invariant optimal control theory and differential geometry. II. Jacobi curves for singular extremals”, J. Dyn. Control Syst., 4 (1998), 583–604 | DOI | MR | Zbl

[3] Bryant R. L., Chern S. S., Gardner R. B., Goldschmidt H. L., Griffiths P. A., Exterior differential systems, Mathematical Sciences Research Institute Publications, 18, Springer-Verlag, New York, 1991 | MR | Zbl

[4] Bryant R. L., Hsu L., “Rigidity of integral curves of rank 2 distributions”, Invent. Math., 114 (1993), 435–461 | DOI | MR | Zbl

[5] Cartan É., “Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre”, Ann. Sci. École Norm. Sup. (3), 27 (1910), 109–192 | MR | Zbl

[6] Cartan É., “Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes”, Bull. Soc. Math. France, 42 (1914), 12–48 | MR

[7] Engel F., “Zur Invariantentheorie der Systeme von Pfaff'schen Gleichungen”, Berichte Ges. Leipzig Math.-Phys. Classe, 41 (1889), 157–176

[8] Jean F., “The car with $n$ trailers: characterisation of the singular configurations”, ESAIM Contrôle Optim. Calc. Var., 1 (1996), 241–266 | DOI | MR | Zbl

[9] Krasil'shchik I. S., Lychagin V. V., Vinogradov A. M., Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, 1, Gordon and Breach Science Publishers, New York, 1986 | MR

[10] Kumpera A., Rubin J. L., “Multi-flag systems and ordinary differential equations”, Nagoya Math. J., 166 (2002), 1–27 | MR | Zbl

[11] Kumpera A., Ruiz C., “Sur l'équivalence locale des systèmes de Pfaff en drapeau”, Monge–Ampère Equations and Related Topics, eds. F. Gherardelli, Ist. Naz. Alta Math. F. Severi, Rome, 1982, 201–248 | MR

[12] Montgomery R., Zhitomirskii M., “Geometric approach to Goursat flags”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 459–493 | DOI | MR | Zbl

[13] Montgomery R., Zhitomirskii M., Points and curves in the monster tower, Mem. Amer. Math. Soc. (to appear) | MR

[14] Mormul P., “Multi-dimensional Cartan prolongation and special $k$-flags”, Geometric Singularity Theory, Banach Center Publ., 65, eds. H. Hironaka, S. Janeczko and S. Lojasiewicz, Polish Acad. Sci., Warsaw, 2004, 157–178 ; also available at http://www.mimuw.edu.pl/~mormul/special.pdf | MR | Zbl

[15] Mormul P., “Geometric singularity classes for special $k$-flags, $k\ge2$, of arbitrary length”, Singularity Theory Seminar, 8, ed. S. Janeczko, Warsaw Univ. of Technology, 2003, 87–100

[16] Mormul P., “Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors”, Cent. Eur. J. Math., 2 (2004), 859–883 | DOI | MR | Zbl

[17] Mormul P., “Special 2-flags, singularity classes, and polynomial normal forms for them”, Sovrem. Mat. Prilozh., 33 (2005), 131–145 (in Russian) | MR

[18] Pasillas-Lépine W., Respondek W., “Contact systems and corank one involutive subdistributions”, Acta Appl. Math., 69 (2001), 105–128 ; math.DG/0004124 | DOI | MR | Zbl

[19] Shibuya K., Yamaguchi K., “Drapeau theorem for differential systems”, Diff. Geom. Appl. (to appear)

[20] Tanaka N., “On differential systems, graded Lie algebras and pseudogroups”, J. Math. Kyoto Univ., 10 (1970), 1–82 | MR | Zbl

[21] von Weber E., “Zur Invariantentheorie der Systeme Pfaff'scher Gleichungen”, Berichte Ges. Leipzig Math.-Phys. Classe, 50 (1898), 207–229 | Zbl

[22] Yamaguchi K., “Contact geometry of higher order”, Japan. J. Math. (N.S.), 8 (1982), 109–176 | MR | Zbl

[23] Yamaguchi K., “Geometrization of jet bundles”, Hokkaido Math. J., 12 (1983), 27–40 | MR | Zbl

[24] Zelenko I., “Fundamental form and the Cartan tensor of $2,5$ distributions coincide”, J. Dynam. Control Syst., 12 (2006), 247–276 | DOI | MR | Zbl