42 research outputs found
Variational Approach to Differential Invariants of Rank 2 Vector Distributions
In the present paper we construct differential invariants for generic rank 2
vector distributions on n-dimensional manifold. In the case n=5 (the first case
containing functional parameters) E. Cartan found in 1910 the covariant
fourth-order tensor invariant for such distributions, using his
"reduction-prolongation" procedure. After Cartan's work the following questions
remained open: first the geometric reason for existence of Cartan's tensor was
not clear; secondly it was not clear how to generalize this tensor to other
classes of distributions; finally there were no explicit formulas for
computation of Cartan's tensor. Our paper is the first in the series of papers,
where we develop an alternative approach, which gives the answers to the
questions mentioned above. It is based on the investigation of dynamics of the
field of so-called abnormal extremals (singular curves) of rank 2 distribution
and on the general theory of unparametrized curves in the Lagrange
Grassmannian, developed in our previous works with A. Agrachev . In this way we
construct the fundamental form and the projective Ricci curvature of rank 2
vector distributions for arbitrary n greater than 4.
For n=5 we give an explicit method for computation of these invariants and
demonstrate it on several examples. In our next paper we show that in the case
n=5 our fundamental form coincides with Cartan's tensor.Comment: 37 pages, SISSA preprint, 12/2004/M, February 2004, minor corrections
of misprint
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
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
A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class
In 1910 E. Cartan constructed the canonical frame and found the most
symmetric case for maximally nonholonomic rank 2 distributions on a
5-dimensional manifold. We solve the analogous problems for rank 2
distributions on an n-dimensional manifold for arbitrary n greater than 5. Our
method is a kind of symplectification of the problem and it is completely
different from the Cartan method of equivalence.Comment: 8 page
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