# 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 $f\colon(-1,1)\to X,$ 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 $\gamma\subset\mathbb{C}$ is said to be an analytic curve (or analytic arc) if every point of $\gamma$ has an open neighbourhood $\Delta$ for which there is an onto conformal map $f\colon{\mathbb{D}}\to\Delta$ (where ${\mathbb{D}}\subset\mathbb{C}$ is the unit disc) such that ${\mathbb{D}}\cap{\mathbb{R}}$ is mapped onto $\Delta\cap\gamma$ 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 ${\mathbb{C}}^{2}$ 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 AnalyticCurve 2013-03-22 14:18:03 2013-03-22 14:18:03 jirka (4157) jirka (4157) 7 jirka (4157) Definition msc 30-00 msc 54-00 analytic arc smooth analytic curve real analytic curve FreeAnalyticBoundaryArc