(in Polish) Osobliwości wybranych dystrybucji geometrycznych 1000-1M22ODG
Definition of an abstract differentiable manifold. Various equivalent definitions of the tangent bundle to a manifold. Example - matrix manifolds O(n) in R^{n*n}, description of theit tangent bundles.
Vector fields on manifolds - smooth sections in tangent bundles. The Lie bracket of vector fields - definition, basic properties and examples, including on the manifolds O(n).
Geometric distribution, i.e., subbundle in the tangent bundle. Two mutually dual descriptions: in the language of vector fields and in those of differential 1-forms. Involutive distributions and Frobenius theorem (1877). Contact structures on odd-dimensional manifolds and Darboux theorem (1882).
Rank-2 distributions on 4-dimensional manifolds. Engel theorem (1889). Weak derived (or Lie) flag of a distribution, and derived flag of a distribution. Small ('sgrv' in brief) and big ('bgrv' in brief) growth vectors of a distribution at a point. Nagano's and Sussmann's theorems about orbits. Bracket-generating (or completely nonholonomic) distributions. The nonholonomy degree of a distribution at a point (= the length of the sgrv at a point). Theorem (only Vermutung, as a matter of fact) of von Weber (1898).
Local equivalence of distributions. A famous counterexample (Kumpera et al, 1978) to von Weber's Vertmutung.
Not numerous stable pairs (distribution's rank, manifold's dimension), (2,4) among them. Historically mportant unstable pairs (2,5) and (3,5) (Cartan, 1910). Absolute equivalence of Cartan (1914).
The growth condition of Zhitomirskii (1990) and the local description of the underlying distributions.
The growth condition of von Weber, nowadays: Goursat condition. Kinematical interpretation of Goursat distributions (i.e., those varifying the Goursat condition): a simplified car towing a string of passive trailers (car + trailers). Jean's recurrencies (1996) for the sgrv's of Goursat distributions (so also for their nonholonomy degrees). Re-phrasing of Jean's stratification/car+trailers model in the singularity theory language.
Cartan prolongation of rank-2 distributions (Cartan 1914, Bryant-Hsu 1993).
Its main application - the Goursat Monster Tower (GMT, Montgomery-Zhitomirskii 2001). The RTV stratification of the stages of the GMT. Information on the local classification of singularities of Goursat structures up to the stage No 8 inclusively. Close relationship between Goursat singularities and singularities of Legendre curves tangent to a fixed contact structure in 3D (Ishikawa 2002, Montgomery-Zhitomirskii 2010).
Nilpotent Lie algebras. Theorem (M., 2000) that Goursat distributions are (effectively!) locally nilpotentizable. Recursive formulas for the nilpotency degrees of the nilpotent algebras, called Kumpera-Ruiz, that locally express the Goursat distributions. Comparison with Jean's recurrencies for the nonholonomy degrees.
ATTENTION. Something, in general, very different from nilpotent algebra locally expressing a distribution, is the nilpotent approximation, around a given point, of an arbitrary completely nonholonomic distribution. DEFINITION of nilpotent approximation of a [germ of] completely nonholonomic distribution.
Motivations underlying the nilpotent approximations, and applications of the latter in robotics and geometric control theory.
Weak (= clasical, mentioned above) and strong (M., 2000) nilpotency of [a germ of] a geometric distribution.
Open problems (local and one global) concerning Goursat distributions. In particular - how rare is the strong nilpotency.
Cartan prolongation of distributions of arbitrary rank. Multi-dimensional monster-type towers and distributions living on their stages (so-called special multi-flags). Stratification of the stages of multi-flags' towers which generalizes the RVT stratification for 1-flags. Information on the same towers (!) known since long in algebraic geometry, and on completely different stratifications of their stages. Local clasification of singularities of special 2-flags on stages up to the fourth
stage. Open questions concerning the stages No 5 and 6. A continuous invariant living on the stage No 7.
Kinematical model of distributions generating special 2-flags (an ideal spececraft dragging in the weightlessness in the 3D space a string of passive satellites).
Special multi-flags are weakly (i.e., classically) nilpotent. The nilpotency degrees of the underlying Lie algebras are effectively (recursively) computable.
Subjects for possible further research (for inst. within a Master Thesis framework) in the nonholonomic analysis sensu largo.
Main fields of studies for MISMaP
mathematics
Type of course
Mode
Assessment criteria
There is no written colloquium during this semestral course. Recitations consist in better illustrating the notions and constructions being introduced and/or used in the lectures, as well as in more careful computations being invoiced only during lectures.
The passing of this course (obtaining a grade ranging from 2 to 5) is on the basis of short essays - studies made by the participants in a non-supervised mode shortly after the end of the semester.
Bibliography
A. Bellaiche, J-J. Risler (eds); Subriemannian Geometry. Birkhauser 1996.
R. Montgomery; A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS 2002.
P. Mormul; Multi-dimensional Cartan prolongation and special k-flags. Banach Center Publications 65 (2004).
A. Agrachev, Yu. Sachkov; Control Theory from the Geometric Viewpoint. Springer 2005.
R. Mongomery, M. Zhitomirskii; Points and Curves in the Monster Tower. Memoirs of the AMS 956 (2010).
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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