2-Dimensional Geometries| 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 Vyy − Vxy2) / (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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send feedback and suggestions to bombelli at olemiss.edu – modified 15 feb 2016