Any two-dimensional (Euclidean) space is locally conformal to flat space; that is, given a two-dimensional manifold $M$ equipped with a (Euclidean) metric $g_{ab}$, it is always possible to locally find a function $\sigma$ such that
$g_{ab} = e^{2\sigma} \delta_{ab}$,
where $\delta_{ab}$ is the metric of flat Euclidean space.
Of course, this is not true globally; for example, if $(M,g_{ab})$ is the round two-sphere, then the function $\sigma$ above must blow up somewhere. Intuitively, this must be the case because the plane and the two-sphere have different topologies. My question is basically whether this topological obstruction is the only problem: that is, if $M$ is a topological two-sphere, is it always possible to find a globally well-behaved function $\sigma$ such that
$g_{ab} = e^{2\sigma} g^\mathrm{(round)}_{ab}$,
where $g^\mathrm{(round)}_{ab}$ is the usual round metric on the two-sphere?
(More generally, is it always possible to find a globally well-behaved $\sigma$ such that $g_{ab} = e^{2\sigma} g^\mathrm{(sym)}_{ab}$, where $g^\mathrm{(sym)}_{ab}$ is the maximally symmetric metric on $M$?)