The theorem of Narasimhan-Seshadri is classically phrased for Riemann surfaces of genus bigger than one.

It then characterises holomorphic bundles which are induced by unitary representations of the fundamental group. Donaldson's theorem on the existence of a Hermitian Einstein metric (which gives a different proof) does not contain the genus condition.

Does anyone know why the "genus >1" condition arise? (or was it just laziness because Grothendieck and Atiyah had aldready covered vector bundles over the Riemann sphere and the torus?)