On variational approach to differential invariants of rank two distributions

We construct differential invariants for generic rank 2 vector distributions on n-dimensional manifolds. where n >= 5. Our method for the construction of invariants is completely different from the Cartan reduction-prolongation procedure. it is based on the dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the theory of unparameterized curves in the Lagrange Grassmannian, developed in [A. Agrachev, I. Zelenko, Geometry of Jacobi curves I, J. Dynam. Control Syst. 8 (1) (2002) 93-140; II, 8 (2) (2002) 167-215]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n >= 5. In the next paper [I. Zelenko, Fundamental form and Cartan's tensor of (2,5)distributions coincide, J. Dynam. Control. Syst., in press, SISSA preprint, Ref. 13/2004/M, February 2004, math.DG/0402195] we show that in the case n = 5 our fundamental form coincides with the Cartan covariant biquadratic binary form, constructed in 19 10 in [E. Cartan, Les systemes de PUT a cinque variables et les equations aux derivees partielles du second ordre. Ann. Sci. Ecole Normale 27 (3) (1910) 109-192; reprinted in: Oeuvres completes. Partie II, vol. 2, Gautier-Villars, Paris, 1953, pp. 927-1010]. Therefore first our approach gives a new geometric explanation for the existence of the Cartan form in terms of an invariant degree four differential on an unparameterized curve in Lagrange Grassmannians. Secondly, our fundamental form provides a natural generalization of the Cartan form to the cases n > 5. Somewhat surprisingly, this generalization yields a rational function on the fibers of the appropriate vector bundle, as opposed to the polynomial function occurring when n = 5. For n = 5 we give an explicit method for computing our invariants and demonstrate the method on several examples. (c) 2005 Elsevier B.V. All rights reserved.