In this talk we shall discuss some linkages between the PDEs of fluid dynamics and the Gauss-Codazzi-Ricci equations for isometric embeddings of Riemannian manifolds. First, we prove the existence of isometric embeddings of certain negatively curved 2-dimensional surfaces into R³ via a ''fluid dynamic formulation'' of the Gauss-Codazzi equations. The key technique is the method of compensated compactness, previously used by Lax, DiPerna, Morawetz and others to show the existence of solutions to hyperbolic conservation laws and transonic gas dynamics. Second, we give global and intrinsic proofs for the weak rigidity of isometric embeddings of Riemannian/semi-Riemannian manifolds, using the generalised compensated compactness theorems recently established in the geometric settings. Third, we discuss an elementary proof for the existence of infinitely many ''wild'' solutions to the Euler equations, previously constructed by De Lellis, Szekelyhidi and others via convex integrations. Our proof relies on a direct dynamic analogue with the isometric embeddings and the celebrated results by Nash and Gromov. The talk is based on joint works with G.-Q. Chen (Oxford), M. Slemrod (Wisconsin-Madison), Dehua Wang (Pittsburgh), and Amit Acharya (Carnegie Mellon).