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SIGMA 5 (2009), 102, 22 pages arXiv:0911.2667
https://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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