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 =  ²(,) (d² d²) ,

where and 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 g's).
* 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

R_abcd = R g_a[c g_d]b .

* Gaussian curvature: For a surface V(x,y),

K = (V_xx V_yy – V_xy²) / (1 + V_xx²V_yy²)² .

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 = d² + f ²() d² ,

and the only nonvanishing connection coefficients in these coordinates are

¹₂₂ = f f ' ,   ²₁₂ = ²₂₁ = 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-in [constant negative Gaussian curvature].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 11 jun 2008