We study a rotating and expanding, Godel type metric, originally considered by Korotkii and Obukhov, showing that, in the limit of large times and nearby distances, it reduces to the open metric of Friedmann. In the epochs when radiation or dust matter dominate the energy density, our solutions are similar to the isotropic ones and, in what concerns processes occurring at small times, the rotation leads only to higher order corrections. At large times, the solution is dominated by a decaying positive cosmological term, with negative pressure, and necessarily describes a quasi-flat universe if the energy conditions have to be satisfied. The absence of closed time-like curves requires a superior limit for the global angular velocity, which appears as a natural explanation for the observed smallness of the present rotation. The conclusion is that the introduction of a global rotation, in addition to be compatible with observation, can enrich the standard model of the Universe, explaining issues like the origin of galaxies rotation and the quasi-flatness problem.