How to find maximal interval of existence for $\sigma$ and show this

This question is from my assignment on Integral Curves and I need help in solving this:

Let $\sigma$ be an integral curve for a vector field X on a manifold M. If $\sigma(0)= \sigma(T)$ for some T>0, show that the maximal interval of existence for $\sigma$ is $ \mathbb{R}$ and that $\sigma(s) =\sigma(s+T)$ for all $s\in \mathbb{R}$.

I don't think I have studied a result which mentions interval of existence and I checked class notes and I couldn't find it. So, I am completely struck on the problem. I am really sorry but I can't provide anything as an attempt.

Kindly consider helping me!

The notion of "maximal interval" comes from the usual study of ODEs, and it comes from the fact that every nice ODE (in particular, the one defining the integral curve of a vector field on a manifold) can be solved locally around a point $t_0 \in \mathbb{R}$.

The maximal interval of existence is the biggest (let's say open) interval containing $t_0$ on which your solution is defined/valid. This might be a bit vague, but it is often clear from context. For example, when solving the ODE $x'(t) = x(t)^2$ with initial condition $x(0) = 1$, your solution will be $x(t) = \dfrac{1}{1-t}$. Hence, the maximal interval of existence (around $t_0 = 0$) is $(-\infty, 1)$, because it blows up at $1$.

Now, try to think what happens (still in the case of a one-dimensional differential equation), what happens when your solution $x$ satisfies $x(0) = x(T)$ for some $T$. Is there anything you can say about its maximal interval of existence?

If you can answer that question, then the answer to your problem on a manifold follows :)

Edit: to clarify what I meant by "every nice ODE can be solved locally", the Picard-Lindelöf theorem states that every ODE of the form $\dot{x}(t) = f(t, x)$ where $f$ is continuous and locally Lipschitz-continuous (for example, this is true when $f$ is $\mathcal{C}^1$) can be solved locally for any initial conditions $t_0, x(t_0)$. In particular, this means you can freely assume that the maximal interval exists and is non-empty :).