Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad.)


Then cover $M$ by open sets $\cup\_iU\_i=M$. 
In a local coordinate chart, $(U\_i,\phi\_i)$, where $\phi\_i:U\_i\rightarrow \mathbb{R}^n$, let us denote these local coordinates by $(\sigma^1,\dots,\sigma^n)$. 


**My question is:** *under what conditions do coordinates such as $(\sigma^1,\dots,\sigma^n)$ exist that cover the entire manifold, $M$, and more importantly why?*


Related questions are: what is the obstruction to extending the local chart to cover the entire surface (except possibly for a discrete set of points in $M$)? Is there a general reasoning that applies to all cases (at least for the case of an orientable compact Riemann surface)?


Any help much appreciated!



non-existence of global coordinates
One particular condition, to give some examples, arises from the [Cartan-Hadamard theorem](https://en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_theorem): if a simply connected manifold admits a complete metric of nonpositive sectional curvature, then it is diffeomorphic to a ball, so admits global coordinates.


Another example: every simply connected surface which is not compact admits a complete metric of constant curvature, and so is diffeomorphic to a disk.


For more complicated surfaces, no finite number of punctures suffices to make a compact surface of genus 1 or more become diffeomorphic to an open subset of the plane, as the homology generators of any open subset of the plane do not intersect one another. Also, the surface clearly must be oriented to be diffeomorphic to an open subset of the plane. On the other hand, an oriented compact surface of genus zero is a sphere, so one puncture turns it into the plane.


