A remark on a theorem of Narasimhan and Ramanan

In this short note, we provide an alternative proof of a notable theorem by Narasimhan and Ramanan. The theorem states that the moduli space of S-equivalence classes of semistable rank 2 vector bundles over a curve X of genus 2 with trivial determinant is isomorphic to \(\mathbb {P}^3\) ${\mathbb{P}}^3$ . Our proof relies on a criterion by Bauer and Szemberg, which characterizes projective spaces among smooth Fano varieties using Seshadri constants.