2-Manifolds| 2-Manifolds |
In General
* Applications: They
are receiving a lot of attention from the mid 1980s with the advent of string
theory.
* 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: 
1(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, 
(M2g)
= 2 – 2 g; In the non-orientable case, (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 =
2.
* 2-torus T2:
Genus g =
1, non-orientable, Euler number =
1.
* 2D projective plane P2:
Genus g =
0, orientable, Euler number =
2.
* Klein
bottle: The "twisted torus" or "curled Möbius
strip" S1 × S1;
Cannot
be imbedded
in
R3 without intersecting itself.
* 2D projective sphere C2:
Genus g = 3, non-orientable, Euler number =
–1.
@ Immersions: Nowik T&A(07) [non-orientable, in R3, 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, 
:= T(p,
0)
/ 
p,
with T(p, 0)
= Teichmüller space, p:=
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
compact];
Weyl
64; Gunning 66; Farkas & Kra 81; Forster 81; Schlichenmaier 89.
@ Related topics: Schaller BAMS(98)
[closed geodesics]; Teschner ht/03-in
[quantization].
References > s.a. tilings [combinatorial curvature].
@ Topology: in Alexandroff 61; Gramain 71.
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send feedback and suggestions to bombelli at olemiss.edu – modified 27
sep
2009