2manifold

A 2-manifold, or surface, is a topological manifold of dimension two. Formally, it is a Hausdorff and second-countable space in which every point has a neighborhood homeomorphic to an open disk in the plane R^2. Consequently, each small patch looks like the plane, while the overall shape can be more complex.

Surfaces may be classified as orientable or non-orientable. An orientable surface has a consistent choice of orientation for tangent spaces, while a non-orientable surface does not. Classic examples include the sphere and torus (both orientable) and the real projective plane and Klein bottle (non-orientable). Other important examples are the plane, the cylinder, and the annulus, which are non-compact or have boundary.

A central result in the theory of 2-manifolds is the classification of closed surfaces. Every connected, compact surface without boundary is homeomorphic either to an orientable surface of genus g (a sphere with g handles) or to a non-orientable surface of genus k (a sphere with k crosscaps). The Euler characteristic χ encodes this: for orientable surfaces, χ = 2 − 2g; for non-orientable surfaces, χ = 2 − k. Thus the torus has χ = 0 (g = 1), the sphere χ = 2 (g = 0), the real projective plane χ = 1 (k = 1), and the Klein bottle χ = 0 (k = 2).

Surfaces with boundary extend this classification: a compact orientable surface with g handles and b boundary components has χ = 2 − 2g − b; a compact non-orientable surface with k crosscaps and b boundaries has χ = 2 − k − b.

Beyond topology, 2-manifolds can carry additional structures, such as smooth or Riemannian structures, connecting them to geometry and complex analysis, where orientable surfaces correspond to Riemann surfaces.