Actions of \({\mathbb{R}}^ p\) on closed manifolds
The authors associate to a non-singular \(C^ r\) \((r\geq 2)\) action F of \({\mathbb{R}}^ p \)on a closed, orientable, connected m-manifold M a symmetric \((q+1)\)-linear mapping \(\alpha_ F: {\mathbb{R}}^ p\times...\times {\mathbb{R}}^ p\to H_{deR}^{2q+1}(M)\), where \(q=m-p\), called the characteristic mapping of F. They study actions through their characteristic mappings. Denote by \(M_ n\) the manifold obtained from \(T^ 2\times [0,1]\) by identifying (x,1) with (A(x),0), where A is the matrix given by \(a_{11}=a_{22}=1\), \(a_{12}=n\) and \(a_{21}=0.\) The main theorem of the paper is the following: If F is an \({\mathbb{R}}^ 2\)-action on \(M_ n\), \(n\neq 0\), such that \(\alpha_ F\) is non- degenerate, then F has a compact orbit and any \({\mathbb{R}}^ 2\)-action G sufficiently \(C^ 1\)-near to F has a compact orbit, too.
scientific article; zbMATH DE number 4056600
scientific article