2-Dimensional
Geometries |

**In General** > s.a. 2D gravity; 2D
manifolds; Geometric Topology.

* __Result__: All metrics are conformally flat, i.e., they can be locally
written as

d*s*^{2} = ± Ω^{2}(*α*, *β*)
(d*α*^{2} ± d*β*^{2})
,

where *α* and *β* are conjugate harmonic coordinates.

* __Riemannian__: There are
3 different kinds of geometry; Given any closed 2-manifold, it can be given
a (unique) metric such that we get one of the following:

* R* > 0: elliptic (S^{2});

* R* = 0: parabolic (T^{2});

* R* < 0:
hyperbolic (all higher orientable *g*s).

* __Lorentzian__: The only compact 2-manifolds which admit a metric of signature
(–, +) are the 2-torus T^{2} and the Klein bottle (thus, e.g., S^{2} does not
admit a Lorentzian metric).

* __Curvature__: The Einstein
tensor vanishes identically, thus ∫_{M}*R* d*v* can only be a topological term plus a surface term (> see
the gauss-bonnet theorem for the
positive-definite case); The Riemann tensor is given by

*R _{abcd}*
=

* __Gaussian curvature__: For
a surface *z* = *V*(*x*, *y*) in \(\mathbb R\)^{3},

*K* = (*V _{xx} V_{yy}* –

@ __Lorentzian__: Vatandoost & Bahrampour JMP(12) [necessary and sufficient conditions for admitting a continuous sphere order representation]; Kim JGP(15)-a1501 [embeddings into the 2D Einstein universe]; Kim JGP(15)-a1501 [conformal diffeomorphisms and causal automorphisms].

> __Hyperbolic__:
see Wikipedia page.

**Special Metrics** > s.a. Zollfrei
Metric.

* __Constant curvature__: In genus 0, the sphere S^{2}; In genus 1, the
flat
torus T^{2}; In genus 2, the double torus, which can be unfolded into an
octagon
in its universal covering, the hyperbolic space H^{2}.

* __Circular symmetry__: In the Riemannian case, the metric can be written
as

d*s*^{2} = d*χ*^{2}
+ *f*^{ 2}(*χ*)
d*φ*^{2} ,

and the only non-vanishing connection coefficients in these coordinates are

Γ^{1}_{22}
= *f f* *'* , Γ^{2}_{12}
= Γ^{2}_{21}
= *f*^{ –1}*f '* .

* __Darboux spaces__: Two-dimensional
spaces of non-constant curvature.

@ __References__: Kramer & Lorente JPA(02)gq/04 [double
torus]; Gallo JMP(04)gq [from
second-order differential equations]; Grosche PPN(06)qp/04 [path
integrals on Darboux spaces]; Bertotti et al m.HO/05-proc
[constant negative Gaussian curvature].

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send feedback and suggestions to bombelli at olemiss.edu – modified 15
feb 2016