Objectives After a concise statement of the historical origin of topological manifold and Gauss-Krüger projection, this paper expoundes and analyzes the principle of manifold mapping of cartographic projection.

Methods The Earth ellipsoid or sphere is redefined from the perspective of Riemannian manifold and the non-Euclidean geometric characteristics, and the topological relationship with plane and the influence on map projection are analyzed.

Results Based on the principle of manifold mapping, this paper considers that the basic contradiction of map projection (i.e. the contradiction between the earth surface and the map plane) should include two aspects, namely, un-developability and un-homeomorphism, which have impacts on distortions, domains and singular points of map projection, etc. Meanwhile, the correctness and feasibility of the authors' assertions about the principle of manifold mapping are further verified in the definition and the necessary and sufficient conditions of conformal map, etc.

Conclusions This work expands the research idea for studying map projection from the perspective of Riemannian manifold mapping.