We propose a formulation of gravity theory in the form of a field theory in a flat space-time with a number of dimensions greater than four. Configurations of the field under consideration describe the splitting of this space-time into a system of mutually noninteracting four-dimensional surfaces. Each of these surfaces can be considered our four-dimensional space-time. If the theory equations of motion are satisfied, then each surface satisfies the Regge-Teitelboim equations, whose solutions, in particular, are solutions of the Einstein equations. Matter fields then satisfy the standard equations, and their excitations propagate only along the surfaces. The formulation of the gravity theory under consideration could be useful in attempts to quantize it.