Arkiv för Matematik

Volume 58 (2020)

Number 1

Periodic flows with global sections

Pages: 39 – 56



Khadija Ben Rejeb (Institute of Computer Science and Communication Techniques, University of Sousse, Hammam Sousse, Tunisia)


Let $G = \lbrace h_t \; \vert \; t \in \mathbb{R} \rbrace$ be a continuous flow on a connected $n$-manifold $M$. The flow $G$ is said to be strongly reversible by an involution $\tau$ if $h_{-t} = \tau h_t \tau$ for all $t \in \mathbb{R}$, and it is said to be periodic if $h_s = $ identity for some $s \in \mathbb{R}^\ast$. A closed subset $K$ of $M$ is called a global section for $G$ if every orbit $G(x)$ intersects $K$ in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if $G$ is periodic and strongly reversible by a reflection, then $G$ has a global section.


periodic flow, strongly reversible, reflection, global section

2010 Mathematics Subject Classification

37B05, 37B20, 54H20, 57S05, 57S10

Received 23 February 2019

Received revised 14 July 2019

Accepted 1 August 2019

Published 21 July 2022