<p>This paper is basically constituted by two parts. In the first part, we consider an interesting conjecture formulated by Elsner (Ann de Inst Fourier 49(1):303–331, 1999) and provide an affirmative answer to it under a very mild assumption. Then, we use the preceding result to provide a detailed characterization of (singular) complex projective structures defined on Riemann surface orbifolds and giving rise to injective developing maps defined on the monodromy covering of the surface (orbifold) in question. The relevance of these structures stems from several problems involving vector fields with uniform solutions as well as from problems about “simultaneous uniformization” for leaves of foliations by Riemann surfaces. In this paper, we first describe the local structure of the mentioned projective structures and then go on to derive some global “classification” building on previous well known material.</p>