# $n$-torus

The *$n$-torus*, denoted ${T}^{n}$, is a smooth orientable $n$ dimensional manifold which is the product of $n$ 1-spheres, i.e. ${T}^{n}=\underset{n}{\underset{\u23df}{{S}^{1}\times \mathrm{\cdots}\times {S}^{1}}}$.

Equivalently, the $n$-torus can be considered to be ${\mathbb{R}}^{n}$ modulo the action (vector addition) of the integer lattice ${\mathbb{Z}}^{n}$.

The $n$-torus is in addition a topological group^{}. If we think of ${S}^{1}$ as the unit circle in $\u2102$ and ${T}^{n}=\underset{n}{\underset{\u23df}{{S}^{1}\times \mathrm{\cdots}\times {S}^{1}}}$, then ${S}^{1}$ is a topological group and so is ${T}^{n}$ by coordinate-wise multiplication. That is,

$$({z}_{1},{z}_{2},\mathrm{\dots},{z}_{n})\cdot ({w}_{1},{w}_{2},\mathrm{\dots},{w}_{n})=({z}_{1}{w}_{1},{z}_{2}{w}_{2},\mathrm{\dots},{z}_{n}{w}_{n})$$ |

Title | $n$-torus |
---|---|

Canonical name | Ntorus |

Date of creation | 2013-03-22 13:59:58 |

Last modified on | 2013-03-22 13:59:58 |

Owner | ack (3732) |

Last modified by | ack (3732) |

Numerical id | 7 |

Author | ack (3732) |

Entry type | Definition |

Classification | msc 22C05 |

Classification | msc 54B10 |

Related topic | Torus |