Actions of cylinders (Q920447)

Let M be a closed orientable connected m-manifold. Let \({\mathcal C}\) be a foliation of M given by a nonsingular \(C^ r\)-action (r\(\geq 2)\) C of the cylinder \(T^ k\times {\mathbb{R}}^ 1\) on M. Let \(A^ r({\mathbb{R}}^ p,M,{\mathcal C})\), \(p=k+1\), be the space of all nonsingular \(C^ r\)-actions F of \({\mathbb{R}}^ p\) on M with underlying foliation \({\mathcal C}\). The authors prove that if \({\mathcal C}\) has no \(C^ r\) first integrals, then each action \(F\in A^ r({\mathbb{R}}^ p,M,{\mathcal C})\) is linearly equivalent to an action of the cylinder \(T^ k\times {\mathbb{R}}^ l\). As a corollary, they obtain the degeneracy of the characteristic mapping \(\alpha_ F\) of actions \(F\in A^ r({\mathbb{R}}^ p,M,{\mathcal C})\) in the case \(k=m-2\), \(l=1\), where \(\alpha_ F\) was defined in the authors' paper ``Actions of \({\mathbb{R}}^ p\) on closed manifolds'' [Topology Appl. 29, 41-54 (1988; Zbl 0647.57029)]. The relation of concordance in \(A^ r({\mathbb{R}}^ p,M,{\mathcal C})\) is defined and the following theorem is proved: If \(k=m- 2\), \(l=1\) and \({\mathcal C}\) has no \(C^ r\) first integrals, then each action in \(A^ r({\mathbb{R}}^{m-1},M,{\mathcal C})\) is concordant to a linear reparametrization of C.