From the Nash–Kuiper theorem of isometric embeddings to the Euler equations for steady fluid motions: Analogues, examples, and extensions

Direct linkages between regular or irregular isometric embeddings of surfaces and steady compressible or incompressible fluid dynamics are investigated in this paper. For a surface (M, g) isometrically embedded in R3, we construct a mapping that sends the second fundamental form of the embedding to the density, velocity, and pressure of steady fluid flows on (M, g). From a Partial Differential Equations perspective, this mapping sends solutions to the Gauss–Codazzi equations to the steady Euler equations. Several families of special solutions of physical or geometrical significance are studied in detail, including the Chaplygin gas on standard and flat tori as well as the irregular isometric embeddings of the flat torus. We also discuss tentative extensions to multiple dimensions.