This is problem 9-1 from John Lee's Introduction to Smooth Manifolds. Suppose $M$ is a smooth manifold $X$ is a smooth vector field on $M$ and $\gamma$ is a maximal integral curve of $X$.
Show that the image of $\gamma$ is an immersed submanifold of $M$, diffeomorphic to $S^1$ if $\gamma$ is periodic and nonconstant.
If $\gamma$ is periodic and nonconstant, we have the period $T$, and we can consider the smooth covering map $\pi:\mathbb{R} \to S^1$ defined by $\pi(x)=e^{2\pi it/T}$. Since $\gamma$ is constant on the fibers of $\pi$ it descends to a smooth map $\tilde{\gamma}:S^1 \to M$ with $\tilde{\gamma}\circ \pi=\gamma$. Since $\gamma(t)=\gamma(t')$ iff $t-t'=kT$ for some $k\in \mathbb{Z}$, $\tilde{\gamma}$ is injective. Hence, $\tilde{\gamma}$ is an injective smooth immersion and its image is an immersed submanifold of $M$ diffeomorphic to $S^1$ using Proposition 5.18 of the book (Images of Immersions as Submanifolds) which is Suppose $M$ is a smooth manifold with or without boundary, $N$ is a smooth manifold, and $F:N \to M$ is an injective smooth immersion. Let $S=F(N).$ Then $S$ has a unique topology and smooth structure such that it is a smooth submanifold of $M$ and such that $F:N\to S$ is a diffeomorphism onto its image.
My question is how can we say that the image of the original curve $\gamma$ is also an immersed submanifold of $M$ diffeomorphic to $S^1$? $\gamma$ is not injective, even though it is an immersion so I cannot apply the Proposition. I know that its image is the same as the image of $\tilde{\gamma}$, but $\gamma$ has a different domain so I don't think I can just translate results about $\tilde{\gamma}$ to $\gamma$. How can I resolve this issue? I would greatly appreciate some help.
Quote from Lee:
In other words, an immersed submanifold $S$ of $M$ is the image of an injective immersion $i : S' \to M$.
Here is Problem 9-1 :
(a) We say $\gamma$ is periodic if there is a number $T > 0$ such that $\gamma(t + T) = \gamma(t)$ for all $t \in \mathbb R$. Show that exactly one of the following holds:
$\quad$ (1) $\gamma$ is constant.
$\quad$ (2) $\gamma$ is injective.
$\quad$ (3) $\gamma$ is periodic and nonconstant.
(b) Show that if $\gamma$ is periodic and nonconstant, then there exists a unique positive number $T$ (called the period of $\gamma$) such that $\gamma(t) = \gamma(t')$ if and only if $t -t' = kT$ for some $k \in \mathbb Z$.
(c) Show that the image of $\gamma$ is an immersed submanifold of $M$; diffeomorphic to $\mathbb R, S^1$, or $\mathbb R^0$.
It seems that you accepted (b) and proved that if $\gamma$ is periodic and nonconstant, then the image of $\gamma$ is an immersed submanifold of $M$ which is diffeomorphic to $S^1$. Actually the image of $\gamma$ is an embedded submanifold $M$ which is diffeomorphic to $S^1$.
I do not know whether you proved (a), but it is clear that your proof works for case (3).
To prove (a) let us distinguish a number of cases.
$\gamma$ is constant. This is (1) above.
$\gamma$ is non-constant.
Case 2 splits into two subcases:
$\quad$ (i) $\gamma$ is injective. This is (2) above. In this case the image of $\gamma$ is trivially an immersed submanifold.
$\quad$ (ii) $\gamma$ is not injective.
This means that $\gamma : J \to M$ has the property $\gamma(t_1) = \gamma(t_2) =: p$ for some points $t_1 < t_2$. By the translation lemma we may assume that $t_1 = 0$. Then $\gamma$ is the unique maximal integral curve with $\gamma(0) = p$. Now let $J^* = \{ t - t_2 \mid t \in J\}$ and $\gamma^* : J^* \to M, \gamma^*(t) = \gamma(t + t_2)$. By the translation lemma this is again an integral curve. Since $\gamma^1*(0) = \gamma(t_2) = p$, by maximality of $\gamma$ we must have $J^* \subset J$ and $\gamma^* = \gamma \mid_{J^*}$. This is possible only when $J = \mathbb R$ and $\gamma^* = \gamma$. This means that $\gamma$ is periodic with period $t_2$.
Thus we get (3) above.