I don't really understand what this question is asking for...
First of all, A curve $\gamma:\mathbb{R} \rightarrow \mathbb{R}^n$ is said to be T-periodic if $\gamma(t+T)=\gamma(t)$ for all $t \in \mathbb{R}$.
Second of all, if this curve is not constant and $T \neq 0$, then we say this curve is closed.
Where the curve $\gamma$ is assumed to be smooth.
However, this question asks: If $\gamma:\mathbb{R} \rightarrow \mathbb{R}^n$ is a smooth non-constant curve that is T-periodic for some $T >0$,prove that this curve is closed.
What?? Isn't this just a definition? Am i misunderstanding the definition here?
this is a problem in Chapter one of Andrew Pressley's Elementary Differential Geometry. This problem is given in the section where he stated the definition i given above, so i should use the given definition right? but if i used the given definition, then what is the meaning of asking this question?