On local geometry of nonholonomic rank 2 distributions

Details On local geometry of nonholonomic rank 2 distributions On local geometry of nonholonomic rank 2 distributions

2007-03-22

arxiv.org/abs/math/0703662v1

Mathematics

Differential Geometry

21 pages, this is the long version of the short note math.DG/0504319 (the latter was published in C.R. Acad. Sci. Paris, Ser.

Scientific paper

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in $\mathbb R^5$. We solve the analogous problem for germs of generic rank 2 distributions in ${\mathbb R}^n$ 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 $W_D$ 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 $W_D$ with the structure group of all M\"obius transformations, preserving 0.

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