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

ds2 = ± Ω2(α, β) (dα2 ± dβ2) ,

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 (S2);
  R = 0: parabolic (T2);
  R < 0: hyperbolic (all higher orientable gs).
* Lorentzian: The only compact 2-manifolds which admit a metric of signature (−, +) are the 2-torus T2 and the Klein bottle (thus, e.g., S2 does not admit a Lorentzian metric).
* Curvature: The Einstein tensor vanishes identically, thus ∫M R dv 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

Rabcd = R ga[c gd]b .

* Gaussian curvature: For a surface z = V(x, y) in \(\mathbb R\)3,

K = (Vxx VyyVxy2) / (1 + Vxx2 Vyy2)2 .

@ 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 S2; In genus 1, the flat torus T2; In genus 2, the double torus, which can be unfolded into an octagon in its universal covering, the hyperbolic space H2.
* Circular symmetry: In the Riemannian case, the metric can be written as

ds2 = dχ2 + f 2(χ) dφ2 ,

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

Γ122 = f f ' ,   Γ212 = Γ221 = f −1f ' .

* 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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