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Rigidity of webs and families of hypersurfaces. (English) Zbl 0583.57015
Singularities and dynamical systems, Proc. Int. Conf., Heraklion/Greece 1983, North-Holland Math. Stud. 103, 271-283 (1985).
[For the entire collection see Zbl 0547.00033.]
A (p\(+1)\)-web of codimension n of \({\mathbb{R}}^{qn}\) is a \((p+1)\)-tuple of p to p transversal foliations of codimension n of \({\mathbb{R}}^{qn}\). The authors demonstrate that the local smooth classification of such webs, with \(p\geq q\), is the same as the topological one. A one-parameter family of smooth hypersurfaces of \({\mathbb{R}}^ n\) can be studied by considering the image of a foliated manifold by a smooth map. The authors use the theory developed on webs to prove that local topological stability is not a generic property for one-parameter families of hypersurfaces in \({\mathbb{R}}^ n\).
Reviewer: J.G.Timourian

MSC:
57R30 Foliations in differential topology; geometric theory
57R45 Singularities of differentiable mappings in differential topology
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
57R40 Embeddings in differential topology
32Sxx Complex singularities