Let $F:M\to M$ is a smooth map where $M$ is compact connected smooth manifold and $F\circ F=F$ then show that $F(M)$ is a submanifold of $M$.

Asked Jun 23 '22 at 14:40

Active Jun 23 '22 at 20:53

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Given here composition of $F$ with $F$ gives $F$ then to show $F(M)$ is a submanifold of $M$. I was thinking in the way that $F(M)$ should be inverse image of some regular value of some smooth function which needs to constructed using this property of $F$ but unable to proceed in this direction. Any hints regarding how to proceed?

edited Jun 23 '22 at 20:53

Pedro

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asked Jun 23 '22 at 14:40

User11111

I've cleared some comments regarding old versions of the post. Can you double check that your post now reads as intended? – Pedro Jun 23 '22 at 20:53

Let $F:M\to M$ is a smooth map where $M$ is compact connected smooth manifold and $F\circ F=F$ then show that $F(M)$ is a submanifold of $M$.

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