65 research outputs found
On local geometry of nonholonomic rank 2 distributions
In 1910 E. Cartan constructed a canonical frame and found the most symmetric
case for maximally nonholonomic rank 2 distributions in . We solve
the analogous problem for germs of generic rank 2 distributions in 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 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
with the structure group of all M\"obius transformations, preserving 0.Comment: 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. I, Vol. 342, Issue 8
(15 April 2006), 589-59
Equivalence of variational problems of higher order
We show that for n>2 the following equivalence problems are essentially the
same: the equivalence problem for Lagrangians of order n with one dependent and
one independent variable considered up to a contact transformation, a
multiplication by a nonzero constant, and modulo divergence; the equivalence
problem for the special class of rank 2 distributions associated with
underdetermined ODEs z'=f(x,y,y',..., y^{(n)}); the equivalence problem for
variational ODEs of order 2n. This leads to new results such as the fundamental
system of invariants for all these problems and the explicit description of the
maximally symmetric models. The central role in all three equivalence problems
is played by the geometry of self-dual curves in the projective space of odd
dimension up to projective transformations via the linearization procedure
(along the solutions of ODE or abnormal extremals of distributions). More
precisely, we show that an object from one of the three equivalence problem is
maximally symmetric if and only if all curves in projective spaces obtained by
the linearization procedure are rational normal curves.Comment: 20 page
Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds
Motivated by the geometric theory of differential equations and the
variational approach to the equivalence problem for geometric structures on
manifolds, we consider the problem of equivalence for distributions with fixed
submanifolds of flags on each fiber. We call them flag structures. The
construction of the canonical frames for these structures can be given in the
two prolongation steps: the first step, based on our previous works, gives the
canonical bundle of moving frames for the fixed submanifolds of flags on each
fiber and the second step consists of the prolongation of the bundle obtained
in the first step. The bundle obtained in the first step is not as a rule a
principal bundle so that the classical Tanaka prolongation procedure for
filtered structures can not be applied to it. However, under natural
assumptions on submanifolds of flags and on the ambient distribution, this
bundle satisfies a nice weaker property. The main goal of the present paper is
to formalize this property, introducing the so-called quasi-principle frame
bundles, and to generalize the Tanaka prolongation procedure to these bundles.
Applications to the equivalence problems for systems of differential equations
of mixed order, bracket generating distributions, sub-Riemannian and more
general structures on distributions are given.Comment: 49 pages. The Introduction was extended substantially: we demonstrate
how flag structures appear in the geometry of double fibrations and, using
this language, we discuss the motivating examples in more detai
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