A complete surface of Gaussian curvature zero in Euclidean three space is a cylinder (where a cylinder means the surface generated by the lines parallel to a given axis passing through a fixed curve in the subspace perpendicular to the axis; the plane is a cylinder in this sense when the curve is itself a line). This is due to Pogorelov, although it is usually attributed to Hartman-Nirenberg (they attribute it to Pogorelov). It can be found proved as Theorem 1 in W. Massey's [Surfaces of Gaussian curvature zero in Euclidean $3$-space][1] or in part II of Hartman and Nirenberg's *On spherical image maps whose Jacobians do not change sign* (Hartman and Nirenberg prove a more general result, for hypersurfaces in Euclidean space of arbitrary dimension). The papers of Massey and Hartman-Nirenberg contain more detailed results applicable to the incomplete case. See also chapter IX of the English translation *Extrinsic Geometry of Convex Surfaces* of Pogorelov's book.

  [1]: http://projecteuclid.org/euclid.tmj/1178244205