# Natural group action on mapping torus

Let $$(F,g)$$ be a Riemannian manifold. Let $$G:=Iso(F,g)$$ be the isometry group. Let $$M$$ be the mapping torus of some isometry of $$F$$. So we have a bundle $$F \to M \to S^1$$ $$M$$ has Riemannian cover $$F \times \mathbb{R}$$ and there is natural action of $$G \times \mathbb{R}$$ on $$F \times \mathbb{R}$$. If the mapping torus is trivial $$M\cong F \times S^1$$ then this action on the cover descends to a natural action on the mapping torus. What if the mapping torus is nontrivial? When is there a natural action of $$G \times \mathbb{R}$$ on $$M$$? I am especially interested in the case where $$F$$ is Riemannian homogeneous and this action on the mapping torus is transitive. I was inspired to ask this by a claim in this question

https://mathoverflow.net/questions/410547/exact-condition-for-smooth-homogeneous-to-imply-riemannian-homogeneous

and a similar claim in this question

https://mathoverflow.net/questions/413409/mapping-torus-of-orientation-reversing-isometry-of-the-sphere

that there is a natural action of the group $$O_{n+1}(\mathbb{R}) \times \mathbb{R}$$ on the mapping torus of the antipodal map on $$S^n$$.

A sufficient condition is that the isometry $$\varphi$$ is central in $$G$$, since in that case the action of $$G\times\mathbb{R}$$ preserves fibers of the covering. This is what happens for the mapping tori of the antipodal maps on round $$S^n$$.
• What do you mean by $F$ is central in $G$? $F$ is a manifold and $G$ is the isometry group of $F$. Jan 19, 2022 at 21:32
• @IanGershonTeixeira: In this context, centrality means that the (cyclic) group of covering transformations of $F\times {\mathbb R}\to M$ commutes with $G$. Equivalently, if $\phi\in G$ is the isometry of $M$ defining the mapping torus, you want $\phi$ to be contained in the center of $G$. This condition is sufficient and almost necessary. The necessary and sufficient condition is that the subgroup generated by $\phi$ is normal in $G$. Jan 19, 2022 at 22:25
• @MoisheKohan oh wow necessary and sufficient is always better than sufficient. Would you mind making this comment into an answer? It seems even better than Kajelad's already great contribution. Does the action of $G×\mathbb{R}$ on M factors through an action of $G×SO_2(\mathbb{R})$? Also does $F$ homogeneous imply the action is transitive? I'm just double checking transitivity because what you guys say suggests that mapping torus of antipodal map of $S^2$ was erroneously omitted from the classification in academia.edu/9710510/Three-dimensional_homogeneous_spaces Jan 20, 2022 at 0:10
• @IanGershonTeixeira I misread "Some isometry of $F$" as "Some isometry $F$"; edited. Jan 20, 2022 at 1:16
• @Kajelad For the case of the mapping tori of antipodal maps of round $S^n$ is this action faithful? Or does it factor through a compact group like $O_{n+1} \times SO_2$? Jan 21, 2022 at 14:48