On variational approach to differential invariants of rank two distributions

Abstract

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 1910 in [E. Cartan, Les systemes de Pfaff 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.