506 research outputs found
Bridges Between Subriemannian Geometry and Algebraic Geometry
We consider how the problem of determining normal forms for a specific class
of nonholonomic systems leads to various interesting and concrete bridges
between two apparently unrelated themes. Various ideas that traditionally
pertain to the field of algebraic geometry emerge here organically in an
attempt to elucidate the geometric structures underlying a large class of
nonholonomic distributions known as Goursat constraints. Among our new results
is a regularization theorem for curves stated and proved using tools
exclusively from nonholonomic geometry, and a computation of topological
invariants that answer a question on the global topology of our classifying
space. Last but not least we present for the first time some experimental
results connecting the discrete invariants of nonholonomic plane fields such as
the RVT code and the Milnor number of complex plane algebraic curves.Comment: 10 pages, 2 figures, Proceedings of 10th AIMS Conference on Dynamical
Systems, Differential Equations and Applications, Madrid 201
Contact systems and corank one involutive subdistributions
We give necessary and sufficient geometric conditions for a distribution (or
a Pfaffian system) to be locally equivalent to the canonical contact system on
Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of
that class of systems, in particular, the existence of corank one involutive
subdistributions. We also distinguish regular points, at which the system is
equivalent to the canonical contact system, and singular points, at which we
propose a new normal form that generalizes the canonical contact system on
Jn(R,Rm) in a way analogous to that how Kumpera-Ruiz normal form generalizes
the canonical contact system on Jn(R,R), which is also called Goursat normal
form.Comment: LaTeX2e, 29 pages, submitted to Acta applicandae mathematica
Singularity Classes of Special 2-Flags
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
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