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.
Read more about this topic: Classification Of Manifolds
Famous quotes containing the word dimensions:
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)