On variational approach to differential invariants of rank two distributions

Abstract

In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n = 5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his "reduction-prolongation" procedure (see [12]). After Cartan's work the following questions remained open: first the geometric reason for existence of Cartan's tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan's tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in [4], [5]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n ≥ 5. For n = 5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In the next paper [19] we show that in the case n = 5 our fundamental form coincides with Cartan's tensor.

Twistor Space for Rolling Bodies

“…• We note here a theorem of Cartan that the vanishing of A i 's is actually a necessary and sufficient condition for all Weyl tensorC of the respective radii r 1 and r 2 ) rolling on each other 'without slipping or twisting' has been investigated for a while during recent years, see [1,4,5,17]. It is therefore well known that the distribution D v defined by such a system on the configuration space C(S 2 r1 , S 2 r2 ) is integrable if and only if the radii r 1 and r 2 of the balls, are equal.…”

“…Montgomery/I. Zelenko, [5,17], says that if, in addition to r 1 = r 2 , the ratio of the radii is r 1 : r 2 = 3 or r 1 : r 2 = 1 3 , then the (2, 3, 5) distribution D v has the maximal local symmetry in the non-integrable case, in which case the local symmetry group is G 2 . This remarkable observation gives a 'physical' realization of this exceptional Lie group; a realization unnoticed by mathematicians and physicists for more than 100 years, from the year 1894, when E. Cartan and F. Engel, have shown that this group is a symmetry group of a certain rank two distribution in dimension five [8,10].…”

Cartan’s incomplete classification and an explicit ambient metric of holonomy $$\mathrm{G}_2^*$$

“…This family includes appropriate complexifications of the rolling distributions of two real surfaces of different nonzero constant curvature whose curvatures do not have ratio 9:1; for that ratio the rolling distribution is flat. See [1,7,8,10,32] for much more. -(Section 51) There is a single distribution with infinitesimal symmetry algebra isomorphic to so(3, C)⊕(so(2, C) C 2 ).…”

Geometry of Optimal Control Problems and Hamiltonian Systems

“…Of course, these nice things have some value only if we can effectively find Jacobi curves for singular extremals: their definition was too abstract. Fortunately, this is not so hard; see [5] for the explicit expression of Jacobi curves for a wide class of singular extremals and, in particular, for singular curves of rank 2 vector distributions (these last Jacobi curves have found important applications in the geometry of distributions, see [11,14]). …”