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Assume we have a smooth manifold, $M$, of dimension $n$, endowed with a Riemannian metric. (An example of interest is the case when $M$ has n=2, is orientable and compact, i.e. a compact 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 $(\sigma^1,\dots,\sigma^n)$. We can then write down the corresponding line element associated to this chart in terms of these coordinates, $$ ds^2=g_{ab}d\sigma^ad\sigma^b. $$ My question is: under what conditions do coordinates such as $(\sigma^1,\dots,\sigma^n)$ exist that cover the whole manifold, 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!

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  • $\begingroup$ This won‘t work in many situations due to topological restrictions. One case where it works if $M$ admits a riemannian metric of nonnegative sectional curvature and is simply connected. $\endgroup$ – Frieder Jäckel Aug 23 '18 at 9:43

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