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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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