2-Manifolds

In General
* Applications: They have been receiving a lot of attention since the mid 1980s with the advent of string theory.
* Invariants: The Euler characteristic is the only topological invariant of a surface that can be found by integration.
* Classification: A full topological classification of (closed) 2-manifolds is given in terms of the orientability and the genus g (the Euler number can be obtained from these, see below); Orientable ones are spheres M2g with a number g of handles (or holes),

M20 = S2,    M21 = T2,   ...

Closed non-orientable (one-sided) ones are also classified by the genus, and they are the projective plane, the Klein bottle, etc.
* Fundamental group: π1(M2g) has 2g generators ai, bi, with one relation, a1 b1 a1–1 b1–1 ... ag bg ag–1 bg–1 = 1.
* Euler characteristic: For an orientable manifold, χ(M2g) = 2 – 2 g; In the non-orientable case, χ(M2g) = 2 – g.
* Cobordism: Two closed 2-manifolds are cobordant iff they both have even or odd Euler characteristic; Thus, there are 2 cobordism classes.
* Differentiable structure: Any closed 2-manifold has a unique differentiable structure; Thus, two homeomorphic closed 2-manifolds are also diffeomorphic.
* Decidability: The set of compact 2-manifolds is algorithmically decidable (has an algorithmic description).

With Other Structures > s.a. 2D geometries; riemann curvature.
* Teichmüller space: For genus p, T(p, 0):= Mp / Conf(M) × Diff0(M), where Mp is the space of metrics for genus p, is the cover of the moduli space of a compact Riemannian surface of genus p > 1; It has dimension 6p – 6, and a natural metric and complex structure, from which the metric can be recovered; The first formulation is due to Riemann; Example: T(1, 0) is the upper half-plane, and T(1, 0) theory is elliptic function theory.
@ Teichmüller space: Bers in(70); Wheeler in(70); in Beis BLMS(72); Royden 71; Bers BAMS(81); Fock dg/97 [dual]; Chekhov a0710-ln; Kashaev a0810-in [Teichmüller theory and discrete Liouville equation].

Examples and Related Concepts > s.a. Weingarten Matrix.
* 2-sphere S2: Genus g = 0, orientable, Euler number χ = 2.
* 2-torus T2: Genus g = 1, non-orientable, Euler number χ = 1.
* 2D projective plane P2: Genus g = 0, orientable, Euler number χ = 2.
* Klein bottle: The "twisted torus" or "curled Möbius strip" S1 × S1; Cannot be imbedded in $$\mathbb R$$3 without intersecting itself.
* 2D projective sphere C2: Genus g = 3, non-orientable, Euler number χ = –1.
@ Immersions: Nowik T&A(07) [non-orientable, in $$\mathbb R$$3, classification].

Riemann Surface
* Idea: A smooth 2-manifold with a complex structure (for an oriented 2-manifold, this is the same as a conformal structure).
* Moduli space: For a compact Riemann surface, it is the space of parameters that determine its conformal type, $$\cal M$$:= T(p, 0) / Γp, with T(p, 0) = Teichmüller space, Γp:= Diff(M) / Diff0(M); It is a normal complex space.
* Examples: For a surface of genus g > 1, there are 3(g–1) complex parameters.
@ General references: Springer 57; Ahlfors & Sario 63 [good intro; little on the compact case]; Weyl 64; Gunning 66; Farkas & Kra 81; Forster 81; Schlichenmaier 89; Napier & Ramachandran 11; Donaldson 11; Eynard a1805-ln [compact].
@ Related topics: Schaller BAMS(98) [closed geodesics]; Teschner ht/03-proc [quantization].