TY - JOUR
T1 - Fundamental form and Cartan tensor of (2,5)-distributions coincide
JF - J. Dyn. Control Syst. 12 (2006) 247-276
Y1 - 2006
A1 - Igor Zelenko
AB - In our previous paper 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.
UR - http://hdl.handle.net/1963/2187
U1 - 2057
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -