Fundamental form and Cartan's tensor of (2,5)-distributions coincide

Abstract

In our previous paper (see this arxiv math.DG/0402171) for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 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 in 1910, using his ``reduction- prolongation'' procedure (we call this form Cartan's tensor). In the present paper we prove that our fundamental form coincides (up to constant factor -35) with Cartan's tensor. This result explains geometric reason for existence of Cartan's tensor (originally this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for Cartan's tensor in terms of structural functions of any frame naturally adapted to the distribution.