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Rigidity of integral curves of rank 2 distributions. (English) Zbl 0807.58007
Let \(\Omega_ D(p,q)\) be the space consisting of differentiable curves in a manifold \(M\) joining \(p\) to \(q\) and staying tangent to a distribution \(D\). At most of its points, \(\Omega_ D(p,q)\) after being endowed with an appropriate topology, behaves very much like an infinite dimensional manifold. However, there are sometimes special curves \(\gamma \in \Omega_ D(p,q)\) around which the local structure of \(\Omega_ D(p,q)\) is drastically different. The authors show that for “most” distributions \(D\) of rank 2, such special curves always occur. In particular, the authors are interested in so-called non-regular curves \(\gamma\) at which \(\Omega_ D(p,q)\) fails to be a smooth manifold when endowed with the natural \(C'\)-topology, and where the natural candidate for the tangent space \(T_ \gamma \Omega_ D(p,q)\) fails to be the true tangent space. The paper concentrates on the case of rigid \(D\)-curves, that is, points \(\gamma \in \Omega_ D(p,q)\) which are essentially isolated.

MSC:
58D99 Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
58D10 Spaces of embeddings and immersions
58B99 Infinite-dimensional manifolds
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