The Ehresmann connection for foliations with singularities and the global stability of leaves (Q2501647)

The paper under review is devoted to an investigation of Ehresmann connections beyond the setting of regular foliations. Specifically, let \({\mathcal F}\) be a foliation with singularities on a smooth manifold \(M\) and let \({\mathcal M}\) be a generalized distribution on \(M\). We say that \({\mathcal M}\) is transversal to \({\mathcal F}\) if there exists a Riemannian metric on \(M\) such that for all \(x\in M\) we have the orthogonal direct sum decomposition \(T_xM={\mathcal T}_x\oplus{\mathcal M}_x\), where \({\mathcal T}\) stands for the generalized distribution tangent to the maximal leaves of \({\mathcal F}\). Then \({\mathcal M}\) is the horizontal distribution while \({\mathcal T}\) is the vertical one. A vertical-horizontal homotopy is a piecewise smooth map \(H\colon[0,1]\times[0,1]\to M\) such that \(H(\cdot,t)\) is a horizontal curve and \(H(t,\cdot)\) is a vertical one whenever \(t\in[0,1]\), and in this case the pair of curves \((H(\cdot,0),H(0,\cdot))\) is said to be the base of the homotopy~\(H\). Now the generalized distribution \({\mathcal M}\) is called an Ehresmann connection for the foliation \({\mathcal F}\) if any admissible pair (that is, any pair consisting of a horizontal path and a vertical one that start from the same point) constitute the base of some vertical-horizontal homotopy. There also exists the notion of generalized Ehresmann connection, which requires the latter property only for the admissible pairs \((\sigma,h)\) such that the image of \(\sigma\), perhaps without the endpoints, is contained in some leaf of maximal dimension. One of the main results of the paper says that if a singular foliation \({\mathcal F}\) on \(M\) admits a generalized Ehresmann connection \({\mathcal M}\) satisfying a number of conditions including a property of local transversal projectability and some regular compact leaf of \({\mathcal F}\) has a finite fundamental group, then every leaf of \({\mathcal F}\) is compact and has a finite fundamental group. Other theorems concern global stability properties in the case when \({\mathcal M}\) is an Ehresmann connection and are stated in terms of a natural notion of holonomy group which can be introduced in this case and is called \(*{\mathcal M}\)-holonomy group. This notion allows one to obtain a version of the above described result with the fundamental groups replaced by \(*{\mathcal M}\)-holonomy groups. The paper concludes by a section that includes a number of interesting examples of Ehresmann connections for foliations with singularities.