ON AFFINE AND PROJECTIVE STRUCTURES ON R1EMANN SURFACES* By IRWIN KRA in Stony Brook,New York, U.S.A. 1. Introduction and statement of results. Let M 1 denote the group of affine transformations; that is, the group of conformal self-mappings of the complex plane C. Similarly, let M 2 denote the group of MSbius trans- formations; that is, the group of conformal self-mappings of the complex sphere, C t3 {co}. Let G be a discontinuous (Fuchsian) group of M/Sbius transformations acting on D, where D is either C or A= {z~C; Iz[ < 1} The pair (~k,f) is called a v-deformation (v = 1 or 2) of G if (i) ~ is a homo- morphism of G into My, and (ii) f is a local homeomorphism and a mero- morphic (holomorphic, if v = 1) function on D such that (1) f oA=~k(A) of for AeG. Two v-deformations (~bl,fl) and (~2,f2) are v-equivalent if there is an element B ~ M v such that (2) f2=Bof i and $2(A)=Bo~I(A)oB-~ for A~G. If G acts freely on D, then D is the universal covering space of the Riemann surface D/G, and G is isomorphic to the fundamental group of this surface. In this case the v-equivalence classes of v-deformations of G are in one-to-one correspondence with the projective (if v = 2) or affine (v = 1) structures on D/G as defined by Gunning [3]. A v-deformation of G is the geometric realiza- (*) Research partially sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, under contract No. F44620-67--C-0029. 285