Periodic flows with global sections

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.

Keywords

periodic flow, strongly reversible, reflection, global section

2010 Mathematics Subject Classification

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

Full Text (PDF format)

Received 23 February 2019

Received revised 14 July 2019

Accepted 1 August 2019

Published 21 July 2022