Space form
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. == Reduction to generalized crystallography == The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form M n {\displaystyle M^{n}} with curvature K = − 1 {\displaystyle K=-1} is isometric to H n {\displaystyle H^{n}} , hyperbolic space; with curvature K = 0 {\displaystyle K=0} is isometric to R n {\displaystyle R^{n}} , Euclidean n-space; and with curvature K = + 1 {\displaystyle K=+1} is isometric to S n {\displaystyle S^{n}} , the n-dimensional sphere of points distance 1 from the origin in R n + 1 {\displaystyle R^{n+1}} .