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An elementary example of a Lie group is afforded by O(2), the orthogonal groupMathworldPlanetmath in two dimensions. This is the set of transformationsMathworldPlanetmath of the plane which fix the origin and preserve the distance between points. It may be shown that a transform has this property if and only if it is of the form


where M is a 2×2 matrix such that MTM=I. (Such a matrix is called orthogonalPlanetmathPlanetmathPlanetmath.)

It is easy enough to check that this is a group. To see that it is a Lie group, we first need to make sure that it is a manifold. To that end, we will parameterize it. Calling the entries of the matrix a,b,c,d, the condition becomes


which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the following system of equations:

a2+c2 =1
ab+cd =0
b2+d2 =1

The first of these equations can be solved by introducing a parameter θ and writing a=cosθ and c=sinθ. Then the second equation becomes bcosθ+dsinθ=0, which can be solved by introducing a parameter r:

b =-rsinθ
d =rcosθ

Substituting this into the third equation results in r2=1, so r=-1 or r=+1. This means we have two matrices for each value of θ:

(cosθ-sinθsinθcosθ)  (cosθsinθsinθ-cosθ)

Since more than one value of θ will produce the same matrix, we must restrict the range in order to obtain a bona fide coordinate. Thus, we may cover O(2) with an atlas consisting of four neighborhoodsMathworldPlanetmath:


Every element of O(2) must belong to at least one of these neighborhoods. It its trivial to check that the transition functions between overlapping coordinate patches are

Title O(2)
Canonical name O2
Date of creation 2013-03-22 17:57:38
Last modified on 2013-03-22 17:57:38
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Example
Classification msc 22E10
Classification msc 22E15