A.A. Agrachev, I. Zelenko, Nurowski's conformal structures for (2,5)-distributions via dynamics of abnormal extremals

Authors: A.A. Agrachev, I. Zelenko
Title: Nurowski's conformal structures for (2,5)-distributions via dynamics of abnormal extremals

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.

Conference: Proceed. RIMS workshop Developments of Cartan geometry and related mathematical problems


Year: 2006
Pages: 204-218