×

Singular holonomy of singular Riemannian foliations with sections. (English) Zbl 1160.53011

Let \((M,g)\) be a complete Riemannian manifold. A singular Riemannian foliation with sections of \((M,g)\) is a partition \(\mathcal F\) of \(M\) by connected immersed submanifolds (the leaves) such that: a) \(\mathcal F\) is singular, i.e. for each leaf \(L\) and each \(v\in T_{p}L\) there exists a vector field \(X\) tangent to the foliation such that \(X_{p}=v\), b) \(\mathcal F\) is transnormal, i.e. every geodesic that is orthogonal to a leaf at one point remains orthogonal to every leaf it meets, c) for each regular point \(p\), the set \(\Sigma:=\exp_{p}\left(\nu_{p}L_{p}\right)\), called section, is a complete immersed submanifold that meets all the leaves and meets them always perpendicularly.
In this survey, some author’s results about singular Riemannian foliations with sections, regarding in particular the singular holonomy, are reviewed.

MSC:

53C12 Foliations (differential geometric aspects)
57R30 Foliations in differential topology; geometric theory
PDFBibTeX XMLCite
Full Text: arXiv