FACTOID # 6: Michigan is ranked 22nd in land area, but since 41.27% of the state is composed of water, it jumps to 11th place in total area.
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


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, Möbius 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 [14] 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 [1]), 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 [2]).
In the general case the local uniformization theorem implies the existence of a finite resolving system (see [3]).
For (local) uniformization in analytic geometry and in the theory of functions of a complex variable (Riemann surfaces) cf.
  More results at FactBites »

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m