30-00-AnalyticCurve.tex

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\pmtitle{analytic curve}
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\begin{document}
There are several somewhat different definitions of the word \emph{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.

\begin{defn}
Suppose $X$ is a real analytic manifold.
A curve $\gamma \subset X$ is an \emph{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 $\gamma .$ 
\end{defn}

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.

\begin{defn}
A curve $\gamma \subset \mathbb{C}$ is said to be an {\em analytic curve} (or {\em 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 $f.$
\end{defn}

Other words for this concept are \emph{smooth analytic curve}, in which case the word \emph{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 \emph{real analytic curve}.

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

\begin{defn}
An \emph{analytic curve} is a complex analytic subvariety of dimension 1 of a complex manifold.
\end{defn}

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.

\begin{thebibliography}{9}
\bibitem{Gamelin:complex}
Theodore~B.\@ Gamelin.
{\em \PMlinkescapetext{Complex Analysis}}.
Springer-Verlag, New York, New York, 2001.
\bibitem{Whitney:varieties}
Hassler Whitney.
{\em \PMlinkescapetext{Complex Analytic Varieties}}.
Addison-Wesley, Philippines, 1972.
\end{thebibliography}
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\end{document}