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On local geometry of non-holonomic rank 2 distributions. (English) Zbl 1202.58002

An \((l, n)\)-distribution \((l < n)\) on an \(n\)-dimensional manifold \(M\) is a vector distribution \(D\) of rank \(l\) on \(M\), i.e., a subbundle of \(TM\) whose fibers are \(l\)-dimensional. A related problem, dating from the 19th century, is that of the local equivalence of two distributions of rank \(l\). An explicit solution of the equivalence problem for distributions of type (2,5) was given by E. Cartan, who associated them with systems of partial differential equations. The solution involves the construction of a canonical coframe for non-degenerate distributions of the said type.
The present paper is mainly concerned with an extension of Cartan’s result to arbitrary non-degenerate \((2, n)\)-distributions for \(n\geq 6\) based on the construction of appropriate coframes. It is also proved that most symmetric \((2, n)\)-distributions form an equivalence class with \((2n - 1)\)-dimensional symmetry algebra which is no longer semisimple for \(n\geq 6\).
The authors are motivated by ideas of control theory in which vector distributions arise in a natural way as constraints on the velocities of the admissible trajectories. Their approach is based on the symplectification of the problem, a method essentially different from other algebraic approaches presenting various difficulties related to the symbol of the distribution. Here an odd-dimensional submanifold is associated with the distribution \(D\), and the substantial tool derived from it are the characteristic curves foliating the submanifold. Omitting many technical details, we say that, in summary, the equivalence problem now essentially reduces to the geometry of such curves. The authors claim that their method works also for distributions of rank greater than 2, as they have already announced in arXiv:0807.3267v1.

MSC:

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