  The isometric embedding problem is a fundamental problem in differential
geometry. A longstanding problem is considered in this paper to characterize
intrinsic metrics on a two-dimensional Riemannian manifold which can be
realized as isometric immersions into the three-dimensional Euclidean space. A
remarkable connection between gas dynamics and differential geometry is
discussed. It is shown how the fluid dynamics can be used to formulate a
geometry problem. The equations of gas dynamics are first reviewed. Then the
formulation using the fluid dynamic variables in conservation laws of gas
dynamics is presented for the isometric embedding problem in differential
geometry.
