In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature.
A space of constant curvature which is geodesically complete is called space form and the study of space forms is intimately related to generalized crystallography (see the article on space form for more details).
Two space forms are isomorphic if and only if they have the same dimension, their metrics possess the same signature and their sectional curvatures are equal.
Moritz Epple (2003) From Quaternions to Cosmology: Spaces of Constant Curvature ca. 1873 — 1925, invited address to International Congress of Mathematicians