# 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 set
of transformations^{} 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

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

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 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

$$\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)={\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)}^{T}\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {a}^{2}+{c}^{2}\hfill & \hfill ab+cd\hfill \\ \hfill ab+cd\hfill & \hfill {b}^{2}+{d}^{2}\hfill \end{array}\right)$$ |

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 a parameter $\theta $ and writing $a=\mathrm{cos}\theta $ and $c=\mathrm{sin}\theta $. Then the second equation becomes $b\mathrm{cos}\theta +d\mathrm{sin}\theta =0$, which can be solved by introducing a parameter $r$:

$b$ | $=-r\mathrm{sin}\theta $ | ||

$d$ | $=r\mathrm{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 for each value of $\theta $:

$$\left(\begin{array}{cc}\hfill \mathrm{cos}\theta \hfill & \hfill -\mathrm{sin}\theta \hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill \mathrm{cos}\theta \hfill \end{array}\right)\mathit{\hspace{1em}\hspace{1em}}\left(\begin{array}{cc}\hfill \mathrm{cos}\theta \hfill & \hfill \mathrm{sin}\theta \hfill \\ \hfill \mathrm{sin}\theta \hfill & \hfill -\mathrm{cos}\theta \hfill \end{array}\right)$$ |

Since more than one value of $\theta $ 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 neighborhoods^{}:

$$ | ||

$$ | ||

$$ | ||

$$ |

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 |