<p>The isometric immersion problem of two-dimensional Riemannian manifolds in the three-dimensional Euclidean space is a fundamental problem in differential geometry. This problem can be formulated as solving the so-called Gauss–Codazzi system, which is a system of nonlinear mixed-type partial differential equations that is hyperbolic for negative Gauss curvature and is elliptic for positive one. In this paper, we construct a local smooth solution on the region with negative Gauss curvature for the Gauss–Codazzi system by assigning appropriate boundary data on the given zero Gauss curvature curve. To overcome the difficulty caused by the degeneracy of Gaussian curvature, we adopt the fluid dynamic framework and the characteristic decomposition technique to transform the Gauss–Codazzi system into a new degenerate hyperbolic system with transparent singularity-regularity structures. Based on the iteration method, we establish the existence of local smooth solutions for the new system in a weighted metric space in the partial hodograph plane. The smooth solution of the Gauss–Codazzi system is then obtained by returning the solution in terms of partial hodograph variables to the original physical variables.</p>