Abstract
In this paper, we study special multi-flags on manifolds. A special multiflag is a certain nested sequence of subbundles of the tangent bundle which are derived by Lie brackets. A property of a special multi-flag is characterized by the existence of a completely integrable subdistribution of corank one in the largest distribution in the sequence, which is a so-called covariant subdistribution. It is proved that a one-parameter deformation of a special multi-flag on a compact manifold can be described by a family of global diffeomorphisms of the underlying manifold, if the covariant subdistribution of the largest distribution in the flag is preserved.
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References
J. Adachi, Global stability of distributions of higher corank of derived length one, International Mathematics Research Notices no. 49 (2003), 2621–2638.
R. Bryant, Some aspects of the local and global theory of Pfaffian systems, PhD thesis, Univ. North Carolina, Chapel Hill, 1979.
R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt and P. A. Griffiths, Exterior Differential Systems, Mathematical Sciences Research Institute Publications, 18, Springer-Verlag, New York, 1991.
R. L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions, Inventiones Mathematicae 114 (1993), 435–461.
A. Golubev, On the global stability of maximally nonholonomic two-plane fields in four dimensions, International Mathematics Research Notices no. 11 (1997), 523–529.
J. W. Gray, Some global properties of contact structures, Annals of Mathematics (2) 69 (1959), 421–450.
A. Kumpera nad J. L. Rubin, Multi-flag systems and ordinary differential equations, Nagoya Mathematical Journal 166 (2002), 1–27.
A. Kumpera and C. Ruiz, Sur l’équivalence locale des systèmes de Pfaff en drapeau, in Monge-Amp`ere Equations and Related Topics (Florence, 1980), Ist. Naz. Alta Mat. Francesco Severi, Rome, 1982, pp. 201–248.
J. Martinet, Classes caractéristiques des systèmes de Pfaff, in Lecture Notes in Mathematics, 392, Springer, Berlin, 1974, pp. 30–36.
R. Montgomery, Engel deformations and contact structures, in Northern California Symplectic Geometry Seminar, American Mathematical Society Translations, Series 2, 196, Amer. Math. Soc., Providence, RI, 1999, pp. 103–117.
R. Montgomery and M. Zhitomirskiĭ, Geometric approach to Goursat flags, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 18 (2001), 459–493.
P. Mormul, Multi-dimensional Cartan prolongation and special k-flags, in Banach Center Publications, 65, Polish Acad. Sci., Warsaw, 2004, pp. 157–178.
P. Mormul, Special 2-flags, singularity classes, and polynomial normal forms for them, Sovremennaya Matematika i ee Prilozheniya 33 (2005), 131–145.
W. Pasillas-Lépine and W. Respondek, Contact systems and corank one involutive subdistributions, Acta Applicandae Mathematicae 69 (2001), 105–128.
K. Shibuya and K. Yamaguchi, Drapeaux theory for differential systems, Differential Geometry and its Applications, to appear.
K. Yamaguchi, Contact geometry of higher order, Japanese Journal of Mathematics (N.S.) 8 (1982), 109–176.
K. Yamaguchi, Geometrization of jet bundles, Hokkaido Mathematical Journal 12 (1983), 27–40.
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partially supported by 21st Century COE Programs “Mathematics of Nonlinear Structure via Singularity”, “Topological Science and Technology”, Hokkaido University, and Grants-in-Aid for Young Scientists (B), No. 17740027, Ministry of Education, Culture, Sports, Science and Technology, Japan.
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Adachi, J. Global stability of special multi-flags. Isr. J. Math. 179, 29–56 (2010). https://doi.org/10.1007/s11856-010-0072-3
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DOI: https://doi.org/10.1007/s11856-010-0072-3