About periodicity of coordinates given a metric

If I am given a metric how do I decide which coordinate is periodic? Eg. can I look at metric in plane polar coordinates and tell that θ direction is periodic.

Also How do I calculate the period of the periodic coordinate?

what I wanted to ask is related to following question. Given the schwarzschild metric we can transform the coordinates as $\tau$ = i*t and $\rho^2$ = $r - r_h$ (where $r_h$ = horizon radius of black hole ) and show that $\tau$ is periodic if there is no conical singularity at $\rho = 0$, and then find the period of $\tau$. (This period is related to temperature of black hole).

• Given a metric and nothing else, you don't, as Slereah's answer states. Concerning your edit, the absence of the conical singularity is an additional constraint, so this is what forces $\tau$ to be periodic in some specific period. Apr 29, 2016 at 13:32
• In that case how do I find the period of $\tau$ ? Apr 29, 2016 at 17:13
• I spelled it out here. May 2, 2016 at 9:56

$$ds^2 = -dt^2 + d\theta$$