On variational approach to differential invariants of rank two distributions. (English) Zbl 1091.58002

Zelenko, Igor

Differ. Geom. Appl. 24, No. 3, 235-259 (2006).

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.

MSC:

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

Keywords:

nonholonomic distributions; differential invariants; abnormal extremals; Jacobi curves; Lagrange Grassmannian; Cartan form

Citations:

JFM 41.0417.01; Zbl 1045.53051; Zbl 1019.53038; Zbl 0058.08302

References:

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