Classification of Manifolds - Dimensions 2 and 3: Geometrizable

Dimensions 2 and 3: Geometrizable

For more details on this topic, see Surface. For more details on this topic, see 3-manifold.

Every closed 2-dimensional manifold (surface) admits a constant curvature metric, by the uniformization theorem. There are 3 such curvatures (positive, zero, and negative). This is a classical result, and as stated, easy (the full uniformization theorem is subtler). The study of surfaces is deeply connected with complex analysis and algebraic geometry, as every orientable surface can be considered a Riemann surface or complex algebraic curve.

Every closed 3-dimensional manifold can be cut into pieces that are geometrizable, by the geometrization conjecture, and there are 8 such geometries. This is a recent result, and quite difficult. The proof (the Solution of the Poincaré conjecture) is analytic, not topological.

While the classification of surfaces is classical, maps of surfaces is an active area; see below.