Intrinsic Geometry of a Surface

the totality of those geometric properties of a surface that can be obtained only with the aid of surface measurements, without resorting to three-dimensional space (in this case, the distance between two points on the surface is defined as the shortest curves lying on the surface and connecting these points). For example, planimetry studies the intrinsic geometry of a plane, and spherical geometry (arising from the needs of cartography) studies the intrinsic geometry of a sphere. The intrinsic geometry of a deformed surface can be viewed as the geometry of a two-dimensional deformed space. The development of the concept of a deformed space led to G. B. Riemann’s creation of so-called Riemann spaces, which play a large role in contemporary physics.