Journal of Dynamical and Control Systems

, Volume 12, Issue 2, pp 247–276

Fundamental Form and the Cartan Tensor of (2,5)-Distributions Coincide


DOI: 10.1007/s10450-006-0383-1

Cite this article as:
Zelenko, I. J Dyn Control Syst (2006) 12: 247. doi:10.1007/s10450-006-0383-1


In our previous paper, for a generic rank-2 vector distributions on an n-dimensional manifold (n ≥ 5) we constructed a special differential invariant, the fundamental form. In the case n = 5, this differential invariant has the same algebraic nature, as the covariant binary biquadratic form, constructed by E. Cartan, using his “reduction-prolongation” procedure (we call this form the Cartan tensor). In the present paper, we prove that our fundamental form coincides (up to the constant factor −35) with the Cartan tensor. This result explains geometrically the reason for the existence of the Cartan tensor (originally, this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in the Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for the Cartan tensor in terms of the structural functions of any frame naturally adapted to the distribution.

Key words and phrases.

Nonholonomic distributionsPfaffian systemsdifferential invariantsabnormal extremalsJacobi curvesLagrange Grassmannian

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.S.I.S.S.A.TriesteItaly