# spherical metric

Suppose that $\widehat{\u2102}:=\u2102\cup \{\mathrm{\infty}\}$ is the extended complex plane^{} (the Riemann sphere).

###### Definition.

Suppose $\gamma :[0,1]\to \widehat{\u2102}$ is a path in $\widehat{\u2102}$. The spherical length of $\gamma $ is defined as

$$\mathrm{\ell}(\gamma ):=2{\int}_{\gamma}\frac{|dz|}{1+{|z|}^{2}}=2{\int}_{0}^{1}\frac{|{\gamma}^{\prime}(t)|}{1+{|\gamma (t)|}^{2}}\mathit{d}t.$$ |

###### Definition.

Let ${z}_{1},{z}_{2}\in \widehat{\u2102}$, and let $\mathrm{\Gamma}$ be the set of all paths
in $\widehat{\u2102}$ from ${z}_{1}$ to ${z}_{2}$, then the distance^{} from
${z}_{1}$ to ${z}_{2}$ in the spherical metric is defined as

$$\sigma ({z}_{1},{z}_{2}):=\underset{\gamma \in \mathrm{\Gamma}}{inf}\mathrm{\ell}(\gamma ).$$ |

More intuitivelly this is the shortest distance to travel from ${z}_{1}$ to ${z}_{2}$ if we think of these points as being on the Riemann sphere, and we can only travel on the Riemann sphere itself (we cannot “drill” a straight line from ${z}_{1}$ to ${z}_{2}$).

## References

- 1 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.

Title | spherical metric |
---|---|

Canonical name | SphericalMetric |

Date of creation | 2013-03-22 14:18:41 |

Last modified on | 2013-03-22 14:18:41 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 54-00 |

Classification | msc 30A99 |

Defines | spherical length |