A new formulation of string theory that is applicable to curved target spaces will be developed. The observables in the new formulation represent the deformations of the image surface. The path integral is ‘‘gauge fixed,’’ but it retains a global gauge invariance. A measure for the embedding integration is developed which encodes a new geometric regularization procedure. The formulation can be applied to target spaces of any dimension. A new interpretation is given to the critical dimension concept, which leads to the ‘‘dynamics of dimension,’’ whereby mass terms or interactions dynamically appear in the theory. Calculations are presented for the genus zero and genus 1 surfaces. At no point is a cosmological term added to the action, and no conformal anomaly is present. There is some discussion of the connection with general relativity. Certain mathematical aspects arise and are included in the Appendices.
