Suppose $\gamma$ is a maximal integral curve of $X\in\mathfrak{X}(M)$, where $M$ is smooth manifold. I am trying to prove that $\gamma$ is injective or periodic.
Here is my attempt:
Suppose $\gamma$ is not injective. Then there are distinct $t,t'\in\mathbb{R}$ such that $\gamma(t)=\gamma(t')$ and $t<t'$. Let $T=t'-t$. Then $\gamma$ and $\delta$, defined by $\delta(s)=\gamma(s+T)$, are both integral curves starting at $\gamma(t)=\gamma(t')$. By the uniqueness of maximal integral curves, we must have that $\gamma$ is defined on $(t-\varepsilon,t'+T+\varepsilon)$ for some $\varepsilon>0$ and that $\gamma(s)=\gamma(s+T)$ whenever $s$ and $s+T$ are in the domain of $\gamma$. Hence $\gamma$ is periodic with period $T$.
Is this correct?