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Encyclopedia > Uniformization theorem

In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. In fact, one can find a metric with constant Gauss curvature in any given conformal class.

From this, a classification of surfaces follows. A surface is a quotient of one of the following by a free action of a discrete subgroup of an isometry group:

1. the Euclidean plane (curvature 0),
2. the sphere (curvature +1), or
3. the hyperbolic plane (curvature -1)

The first case include all surfaces with zero Euler characteristic: a cylinder, torus, M�bius strip, Klein bottle or Euclidean plane. In the second case we have all surfaces with positive Euler characteristic: only the sphere and projective plane. The last case we have all surfaces with negative Euler characteristic; almost all surfaces are hyperbolic. Results from FactBites:

 Citations: On Thurston's uniformization theorem for three-dimensional manifolds - Morgan (ResearchIndex) (2502 words) Citations: On Thurston's uniformization theorem for three-dimensional manifolds - Morgan (ResearchIndex) Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, in : J.Morgan, H.Bass, The Smith Conjecture, A.P., 1984, 37-126. Theorem 1.1 of  states, among other things, that if is any proper 1 manifold in a compact, connected 3 manifold Y such that meets every 2 sphere in Y at least twice and every projective plane in Y at least once, then is....
 Springer Online Reference Works (295 words) The existence of a resolving system (the local uniformization theorem) was proved for arbitrary varieties over a field of characteristic zero (see ), and also for two-dimensional varieties over any field and three-dimensional varieties over an algebraically closed field of characteristic other than 2, 3 or 5 (see ). In the general case the local uniformization theorem implies the existence of a finite resolving system (see ). For (local) uniformization in analytic geometry and in the theory of functions of a complex variable (Riemann surfaces) cf.
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