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An estimate of the functional dimension of the orbit space of germs of distributions of general position. (Russian) Zbl 0667.58003
The authors consider the orbits of the (obvious) action of the group \(Diff_ n\) of germs of diffeomorphisms of \({\mathbb{R}}^ n\) on the set \(W^ k_ n\) of germs of k-dimensional distributions at \(0\in {\mathbb{R}}^ n\) (i.e. the k-dimensional subbundles of the tangent bundle \(T{\mathbb{R}}^ n)\). They introduce the notion of the (jet) asymptotical dimension and use it to prove that for all (k,n), the orbits of general position have infinite codimension, except for the classical cases - when there exists an open dense orbit. This problem arose in connection with the authors’ work in nonholonomic differential geometry [cf. A. M. Vershik and V. Ya. Gershkovich, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 16, 5-85 (1987)].
Reviewer: D.Repovš

MSC:
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
58A20 Jets in global analysis
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