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 incompressible fluid dynamics are investigated in this paper. For a surface ( M, g) isometrically embedded [Formula: see text], we construct mapping that sends the second fundamental form embedding to density, velocity, pressure flows on g). From Partial Differential Equations perspective, solutions Gauss–Codazzi equations Euler equations. Several families special physical geometrical significance studied detail, including Chaplygin gas standard flat tori as well torus. We also discuss tentative extensions multiple dimensions.