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.

中文翻译：

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全球部分的周期性流动

令$ G = \ lbrace h_t \; \ vert \; t \ in \ mathbb {R} \ rbrace $是相连的$ n $流形$ M $上的连续流。如果$ h _ {-t} = \ tau h_t \ tau $对于$ mathbb {R} $中的所有$ t，则流$ G $可以通过对合$ \ tau $强烈逆转。如果$ h_s = $ \ mathbb {R} ^ \ ast $中某些$ s的身份，则为周期性。如果每个轨道$ G（x）$恰好在一个点相交$ K $，则将$ M $的闭合子集$ K $称为$ G $的全局部分。在本文中，我们研究了“强可逆”和“具有全局截面”这两个属性之间的关系。特别是，我们表明，如果$ G $是周期性的并且通过反射具有很强的可逆性，则$ G $具有全局部分。