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On variational approach to differential invariants of rank two distributions. (English) Zbl 1091.58002
Summary: We construct differential invariants for generic rank 2 vector distributions on \(n\)-dimensional manifolds, where \(n\geq 5\). 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, and I. Zelenko, J. Dyn. Control Syst. 8, No. 1, 93–140 (2002; Zbl 1019.53038); and ibid. 8, No. 2, 167–215 (2002; Zbl 1045.53051)]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary \(n\geq 5\).
In the next paper [I. Zelenko, Fundamental form and Cartan’s tensor of (2,5)-distributions coincide, J. Dyn. 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 [É. Cartan, Ann. Sci. Ecole Norm. 27 (3), 109–192 (1910; JFM 41.0417.01); reprinted in: Œuvres complètes, Partie II, vol. 2, Gauthier-Villars, Paris (1953; Zbl 0058.08302)]. 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.

58A30 Vector distributions (subbundles of the tangent bundles)
53A55 Differential invariants (local theory), geometric objects
58A17 Pfaffian systems
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