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
*M*^{2}_{g}
with a number *g* of handles (or holes),

*M*^{2}_{0} = S^{2}, *M*^{2}_{1} = T^{2}, ...

Closed non-orientable (one-sided) ones are also classified by the genus, and
they are the projective plane, the Klein bottle, etc.

* __Fundamental group__: π_{1}(*M*^{2}_{g})
has 2*g* generators *a*_{i},
*b*_{i}, with one relation,
*a*_{1}* b*_{1}* a*_{1}^{–1}* b*_{1}^{–1 }...
*a*_{g}* b*_{g}* a*_{g}^{–1}* b*_{g}^{–1}
= 1.

* __Euler characteristic__:
For an orientable manifold, *χ*(*M*^{2}_{g})
= 2 – 2 *g*; In the non-orientable case, *χ*(*M*^{2}_{g})
= 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):= *M*_{p} /
Conf(*M*) × Diff_{0}(*M*),
where *M*_{p} 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 6*p* – 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____ S__^{2}:
Genus *g* = 0, orientable, Euler number *χ* = 2.

* __2-torus____ T__^{2}:
Genus *g* = 1, non-orientable, Euler number *χ* = 1.

* __2D projective plane P__^{2}:
Genus *g* = 0, orientable, Euler number *χ* = 2.

* __Klein bottle__: The "twisted torus" or
"curled Möbius strip" S^{1} × S^{1};
Cannot be imbedded in \(\mathbb R\)^{3} without intersecting itself.

* __2D projective sphere C__^{2}:
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) / Γ_{p},
with *T*(*p*, 0)
= Teichmüller space, Γ_{p}:=
Diff(*M*) / Diff_{0}(*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].

> __Online resources__:
see Wikipedia page.

**References** > s.a. tilings [combinatorial curvature].

@ __Topology__: in Alexandroff 61;
Gramain 71;
Wintraecken & Vegter T&IA(13) [topological invariants].

@ __Related topics__: Hoppe & Hynek a1108 [structure constants for certain Lie algebras of vector fields on 2D compact manifolds]

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send feedback and suggestions to bombelli at olemiss.edu – modified 16 may 2018