2-Manifolds| 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].
> 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]
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 16 may 2018