Summary: We present the one and two-loop quantum corrections to the earlier proposed string theory whose world-sheet action measures the linear sizes of the surfaces by the square root of the extrinsic curvature. We find in this model the usual conformal anomaly. Moreover, the one-loop perturbative analysis shows that the dynamics of this model is determined by a reduced number of degrees of freedom compared to the usual string. We point out that this model does not receive any quantum corrections around its classical trajectory. Finally we show that the constraint, relating the induced metric with the string fields, is enforced by radiative corrections and it does not allow the generation of the Polyakov-Kleinert smooth string.