\(C^ 1\) foliations which cannot be approximated by \(C^ 2\) foliations
Let M be a closed manifold and \(\tau\) be a homotopy class of codimension k plane fields on M. Define \(Fol^ r_{\tau}(M)\) to be the space of the codimension k foliations of class \(C^ r\) of M whose tangent bundles are in \(\tau\). In \(Fol^ r_{\tau}(M)\) consider the \(C^ 0\) or \(C^ 1\) topology. The author shows that if dim M\(>2\) and \(\chi (M)=0\), then for any homotopy class \(\tau\) of codimension one plane fields on M, \(Fol^ 2_{\tau}(M)\) is not dense in \(Fol^ 1_{\tau}(M)\) with respect to the \(C^ 0\) topology. If \(\tau\) is a homotopy class of plane fields of codimension greater than one, then \(Fol^ 2_{\tau}(M)\) is note dense in \(Fol^ 1_{\tau}(M)\) with respect to the \(C^ 1\) topology.
approximating \(C^ 2\)-foliations by \(C^ 1\)-foliations
space of the codimension k foliations of class \(C^ r\)
scientific article; zbMATH DE number 4056591