  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 $\mathbb{R}^3$, we
construct a mapping which sends the second fundamental form of the embedding to
the density, velocity, and pressure of steady fluid flows on $(M,g)$. From the
PDE perspectives, 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
multi-dimensions.
