O(2)

still being written

An elementary example of a Lie group is afforded by O(2),the orthogonal group in two dimensions. This is the setof transformations of the plane which fix the origin andpreserve the distance between points. It may be shownthat a transform has this property if and only if it is ofthe form

$$\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)\mapsto M\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right),$$

where $M$ is a $2\times 2$ matrix such that $M^{T}M=I$.(Such a matrix is called orthogonal.)

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

which is equivalent to the following system of equations:

$a^2+c^2$$=1$

$ab+cd$$=0$

$b^2+d^2$$=1$

The first of these equations can be solved by introducing aparameter $\theta$ and writing $a=\cos \theta$ and $c=\sin \theta$. Then the second equation becomes $b\cos \theta +d\sin \theta =0$, which can be solved byintroducing a parameter $r$:

$b$$=-r\sin \theta$

$d$$=r\cos \theta$

Substituting this into the third equation results in $r^2=1$,so $r=-1$ or $r=+1$. This means we have two matrices foreach value of $\theta$:

Since more than one value of $\theta$ will produce the samematrix, we must restrict the range in order to obtain abona fide coordinate. Thus, we may cover $O(2)$ with anatlas consisting of four neighborhoods:

Every element of $O(2)$ must belong to at least one ofthese neighborhoods. It its trivial to check that thetransition functions between overlapping coordinatepatches are