Non-Euclidean geometry: The Gauss formula and an interpretation of partial differential equations (Q1915327)

A number of problems related to a new geometrical approach to the interpretation of differential equations, which is based on regarding them as relations that are generated in some way by special coordinate nets on smooth two-dimensional manifolds with prescribed Gaussian curvature, are discussed. The notion of the \(G\)-class (the Gaussian class) of differential equations, admitting the above-mentioned interpretation, is introduced. The key equality used for developing this idea is the Gauss formula for the curvature of a two-dimensional metric. The prospects of such an approach are based on non-Euclidean geometry in studying nonlinear differential equations.