a space form
Of all Riemannian manifolds, the most symmetric and uniform are those that look the same at every point and in every direction — where curvature is a single constant, never varying. These are the space forms: the curved-space analogues of the perfectly homogeneous, isotropic geometries. They are the geometries in which trigonometry has clean formulas and triangles behave predictably, and they are the foundational models against which all other curvature is compared.
A space form is a complete Riemannian manifold of constant sectional curvature K, meaning K(sigma) takes the same value for every 2-plane sigma at every point. The simply connected ones come in exactly three families, distinguished by the sign of K: the round sphere S^n (K > 0), Euclidean space R^n (K = 0), and hyperbolic space H^n (K < 0), each unique up to scaling and isometry. The classification theorem (Killing-Hopf) states that every complete simply connected Riemannian manifold of constant curvature is isometric to one of these three models. Dropping simple connectivity, general space forms are quotients M-tilde / Gamma of a model M-tilde by a discrete group Gamma of isometries acting freely and properly — for example flat tori, lens spaces, and compact hyperbolic manifolds.
Why they matter: space forms are the comparison spaces of the entire subject — Rauch and Toponogov comparison theorems measure a general manifold against a space form of matching curvature bound. Honest distinctions to keep straight. Constant SECTIONAL curvature is much stronger than Einstein (constant Ricci): every space form is Einstein, but in dimension >= 4 most Einstein manifolds are not space forms. And 'constant curvature' must mean sectional here — a space of constant scalar curvature is enormously more general. The three models are also exactly the geometries whose isometry groups act transitively on frames, making them maximally symmetric.
A flat torus is a space form: take R^2 (curvature 0) and quotient by the lattice Z^2 acting by translations. It is complete, has constant curvature 0 everywhere, but is not simply connected — its fundamental group is Z x Z. So it is a non-simply-connected K = 0 space form, locally indistinguishable from the plane but globally a doughnut.
The flat torus R^2 / Z^2: constant zero curvature, complete, but not simply connected.
'Space form' means constant SECTIONAL curvature, strictly stronger than Einstein (constant Ricci) in dimension >= 4. And the classification into sphere / Euclidean / hyperbolic is for the SIMPLY CONNECTED complete case; general space forms are quotients by discrete isometry groups.