# analytic curve

There are several somewhat different definitions of the word *analytic curve* depending on context.
In the context of a real analytic manifold^{} (for example ${\mathbb{R}}^{n}$), the most generic^{}
definition is perhaps the following.

###### Definition.

Suppose $X$ is a real analytic manifold.
A curve $\gamma \subset X$ is an *analytic curve* if it is a real analytic submanifold
of dimension^{} 1. Equivalently if near each point $p\in \gamma ,$
there exists a real analytic mapping
$f:(-1,1)\to X,$ such that $f$ has nonvanishing differential and maps onto a neighbourhood of $p$ in $\gamma .$

It is sometimes common to equate the mapping $f$ and the curve $\gamma $.
If the curve is as above but instead in the complex plane, we can instead make the following equivalent^{} definition.

###### Definition.

A curve $\gamma \subset \u2102$ is said to be an analytic curve (or analytic arc) if every point of $\gamma $ has an open neighbourhood $\mathrm{\Delta}$ for which there is an
onto conformal map $f:\mathbb{D}\to \mathrm{\Delta}$ (where $\mathbb{D}\subset \u2102$ is
the unit disc^{}) such that $\mathbb{D}\cap \mathbb{R}$ is mapped
onto $\mathrm{\Delta}\cap \gamma $ by $f.$

Other words for this concept are *smooth analytic curve*, in which case the word *analytic curve*
would be reserved for curves with singularities. That is, for real analytic subvarieties of $X$. Some authors will emphasize the fact that this is a real curve and say *real analytic curve*.

In the context of subvarieties the following definition may be used.

###### Definition.

An *analytic curve* is a complex analytic subvariety of dimension 1 of a complex manifold^{}.

Note that locally all complex analytic subvarieties of dimension 1 in ${\u2102}^{2}$ can be parametrized by a the Puiseux parametrization theorem. Perhaps that is why there is the confusion in using the term.

## References

- 1 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.
- 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | analytic curve |
---|---|

Canonical name | AnalyticCurve |

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

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

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 30-00 |

Classification | msc 54-00 |

Synonym | analytic arc |

Synonym | smooth analytic curve |

Synonym | real analytic curve |

Related topic | FreeAnalyticBoundaryArc |