Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 102, 22 pages      arXiv:0911.2667      http://dx.doi.org/10.3842/SIGMA.2009.102
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Singularity Classes of Special 2-Flags

Piotr Mormul
Institute of Mathematics, Warsaw University, 2 Banach Str., 02-097 Warsaw, Poland

Received April 16, 2009, in final form October 30, 2009; Published online November 13, 2009

Abstract
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.

Key words: generalized Cartan prolongation; special multi-flag; special 2-flag; singularity class.

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