Skip Navigation


Journal of the London Mathematical Society Advance Access originally published online on August 30, 2009
Journal of the London Mathematical Society 2009 80(3):545-566; doi:10.1112/jlms/jdp044
This Article
Right arrow FREE Full Text (PDF) Freely available
Right arrow All Versions of this Article:
80/3/545    most recent
jdp044v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Doubrov, B.
Right arrow Articles by Zelenko, I.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2009 London Mathematical Society

On local geometry of non-holonomic rank 2 distributions

B. Doubrov

Department of Applied Mathematics and Computer Science
Belarussian State University
Nezavisimosti Ave. 4
Minsk 220050
Belarus

I. Zelenko

Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
USA
zelenko@math.tamu.edu

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally non-holonomic rank 2 distributions in R5. We solve the analogous problem for germs of generic rank 2 distributions in Rn for n > 5. We use a completely different approach based on the symplectification of the problem. The main idea is to consider a special odd-dimensional submanifold WD of the cotangent bundle associated with any rank 2 distribution D. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. Using the classical theory of curves in projective spaces, we construct the canonical frame of the distribution D on a certain (2n – 1)-dimensional fiber bundle over WD with the structure group of all Möbius transformations, preserving 0. The paper is the detailed exposition of the constructions and the results, announced in the short note (B. Doubrov and I. Zelenko, C. R. Math. Acad. Sci. Paris, Ser. I (8) 342 (2006) 589–594).

2000 Mathematics Subject Classification 58A30, 53A55.

Received November 10, 2007; revised May 27, 2009; published online August 30, 2009.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.