What condition do global coordinates fulfil? This may be a dumb or vague question:
Is there any criterion that a metric tensor needs to fulfill such that coordinates it is expressed in can be called global. Or alternatively what is the definition of global coordinates?
For instance why is one coordinate system for $AdS_n$ called global while others are not?
In particular I am not asking if global coordinates exist but rather how one can check if a set of given coordinates is global. By inspection of the metric tensor or what is the same thing the line element.
 A: The bare minimum requirement for a coordinate chart to be usable in general relativity is essentially that the metric, expressed as a matrix, is always finite and invertible. That is, both $g_{ij}$ and $g^{ij}$ must exist. This requires both that the metric be nondegenerate (which is a coordinate-independent criterion, basically the metric has to have the right signature) and that there are no coordinate singularities when the metric is expressed componentwise in terms of these particular coordinates.
Depending on what we want to do, we will usually want to impose stricter regularity requirements than this. For example, we probably want the metric, expressed in these coordinates, to be such that we can take the derivatives that are required in order to calculate the Riemann tensor -- otherwise we wouldn't be able to state the Einstein field equations.
A valid set of global coordinates is simply a valid coordinate chart that covers all points in the spacetime. There is no requirement that we work in a global chart or that such a chart exist.
A: Basically, this is due to the topology of spacetime. It is also due to how we represent coordinates. 
Take a sphere - for example the earth. If we were to try to represent with the kind of charts we see in books - that is a square - we see we can never do this with just one page - we always need more than one. This is because a square is essentially flat but a sphere is not. 
Were we to instead try to represent it with a globe we see straightaway that only one is neccessary. 
Now, the mathematical apparatus of differential geometry relies on an atlas of patches and these patches are modelled only on the analogue of squares here - Euclidean spaces. It doesn't allow any other kind. 
Thus - if you are modelling spacetime as a sphere - and using standard differential geometry to represent it - your atlas will not be global - in the sense of requiring just one patch - it will always need more than one. 
