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NUROWSKI'S CONFORMAL STRUCTURES FOR (2,5)-DISTRIBUTIONS VIA DYNAMICS OF ABNORMAL EXTREMALS
 

Summary: NUROWSKI'S CONFORMAL STRUCTURES FOR (2,5)-DISTRIBUTIONS
VIA DYNAMICS OF ABNORMAL EXTREMALS
ANDREI AGRACHEV AND IGOR ZELENKO
Abstract. As was shown recently by P. Nurowski, to any rank 2 maximally nonholonomic
vector distribution on a 5-dimensional manifold M one can assign the canonical conformal
structure of signature (3, 2). His construction is based on the properties of the special 12-
dimensional coframe bundle over M, which was distinguished by E. Cartan during his famous
construction of the canonical coframe for this type of distributions on some 14-dimensional
principal bundle over M. The natural question is how "to see" the Nurowski conformal
structure of a (2, 5)-distribution purely geometrically without the preliminary construction
of the canonical frame. We give rather simple answer to this question, using the notion of
abnormal extremals of (2, 5)-distributions and the classical notion of the osculating quadric
for curves in the projective plane. Our method is a particular case of a general procedure
for construction of algebra-geometric structures for a wide class of distributions, which will
be described elsewhere. We also relate the fundamental invariant of (2, 5)-distribution, the
Cartan covariant binary biquadratic form, to the classical Wilczynski invariant of curves in
the projective plane.
1. Introduction
1.1. Statement of the problem. The following rather surprising fact was discovered by
P. Nurowski recently in [8]: any rank 2 maximally nonholonomic vector distribution on a 5-

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics