Locally compact flows on connected manifolds

Abstract

In this paper, we completely characterize locally compact flows G G of homeomorphisms of connected manifolds M M by proving that they are either circle groups or real groups. For M = R m M = \mathbb R^m , we prove that every recurrent element in G G is periodic, and we obtain a generalization of the result of Yang [Hilbert’s fifth problem and related problems on transformation groups, American Mathematical Society, Providence, RI, 1976, pp. 142–146.] by proving that there is no nontrivial locally compact flow on R m \mathbb R^m in which all elements are recurrent.