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 ${ℝ}^n$), the most genericdefinition 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 submanifoldof 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 $\gamma \subset ℂ$ 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 anonto conformal map $f:𝔻\to \mathrm{\Delta }$ (where $𝔻\subset ℂ$ isthe unit disc) such that $𝔻\cap ℝ$ is mappedonto $\mathrm{\Delta }\cap \gamma$ by $f.$

Other words for this concept are smooth analytic curve, in which case the word analytic curvewould 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 ${ℂ}^2$ can be parametrized by a the Puiseux parametrization theorem. Perhaps that is why there is the confusion in using the term.