Metrical distortions are introduced as generalizations of diffeomorphisms as point set transformations within Riemannian geometry. Especially, synthetical coordinates as an explicit Euclidean coordinate system, are derived from the natural affine space over the manifold. As an immediate consequence, Riemann-Lebesgue integration is generalized and a natural integration theory for a special type of partial differential equations is established. As a substantial result, the naming “rubber sheet geometry” for topology is justified.

Riemannian geometry, metrical distortions, synthetical coordinates, geometric partial differential equations.