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Direct linkages between regular or irregular isometric embeddings of surfacesand steady compressible or incompressible fluid dynamics are investigated inthis paper. For a surface $(M,g)$ isometrically embedded in $\mathbb{R}^3$, weconstruct a mapping which sends the second fundamental form of the embedding tothe density, velocity, and pressure of steady fluid flows on $(M,g)$. From thePDE perspectives, this mapping sends solutions to the Gauss--Codazzi equationsto the steady Euler equations. Several families of special solutions ofphysical or geometrical significance are studied in detail, including theChaplygin gas on standard and flat tori, as well as the irregular isometricembeddings of the flat torus. We also discuss tentative extensions tomulti-dimensions.

From the Nash--Kuiper Theorem of Isometric Embeddings to the Euler Equations for Steady Fluid Motions: Analogues, Examples, and Extensions