Singularity Classes of Special 2-Flags

TL;DR

This paper classifies singularity types of special 2-flags, a class of vector distributions generated by generalized Cartan prolongations, providing a geometric understanding of their polynomial normal forms.

Contribution

It constructs a stratification of germs of special 2-flags into singularity classes, giving geometric meaning to their polynomial pseudo-normal forms.

Findings

Stratification of special 2-flags into singularity classes.

Invariant geometric significance for polynomial normal forms.

Rich structure of singularities without functional moduli.

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.