The Special Role of Cartesian Coordinates, Lamé's Method, and the Local Nature of Geometry

We re-examine the role of Cartesian coordinates, the Lamé method, and the principle of general covariance in the understanding of geometry. In Cartesian coordinates, coordinate differentials directly coincide with vector components, while in other orthogonal coordinate systems this is not the case. Lamé’s method introduces Lamé coefficients to define infinitesimal vector components, but its deeper conceptual significance has been historically overlooked. We propose to interpret the Lamé method as a foundation for describing geometry locally through vector differentials, rather than relying exclusively on the postulate of general covariance. In this framework, the metric tensor in Euclidean space always has constant components equal to Kronecker symbols, while in non-Euclidean spaces the dependence of the metric on position should be understood as a local property of geometry. This interpretation clarifies the privileged role of Cartesian coordinates in Euclidean geometry and of Fock’s harmonic coordinates in non-Euclidean geometry. It suggests that the nature of geometry should be determined by local differential structure rather than by the formalism of general covariance.