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