On the orbit structure of \(\mathbb R^n\)-actions on \(n\)-manfolds (Q1773853)

This paper characterizes all non-compact orbits for a locally free \(C^2\)-action of \({\mathbb R}^{n-1}\) on a compact \(n\)-manifold without empty boundary. Specifically speaking, the authors of the paper prove the following result. \textbf{Theorem.} {Let \(\psi\) be a locally free \(C^2\)-action of \({\mathbb R}^{n-1}\) on \(T^{n-1}\times [0,1]\), \(n\geq 2\), tangent to the boundary. If there are no compact orbits in the interior, then all non-compact orbits have the same topological type.} Further, the authors consider \(C^2\)-actions of \({\mathbb R}^n\) on a closed connected orientable real analytic \(n\)-manifold \(N\). They study a kind of \(C^2\)-actions \(\varphi\) of \({\mathbb R}^n\) on \(N\), and show that if \(\varphi\) has a \(T^{n-1}\times{\mathbb R}\)-orbit, then every \(n\)-dimensional orbit is also a \(T^{n-1}\times{\mathbb R}\)-orbit. At the same time, much information on the topology of \(N\) is obtained.