There are several somewhat different definitions of the word analytic curve depending on context. In the context of a real analytic manifold (for example ), the most generic definition is perhaps the following.

Definition 1   Suppose is a real analytic manifold. A curve is an analytic curve if it is a real analytic submanifold of dimension 1. Equivalently if near each point there exists a real analytic mapping such that has nonvanishing differential and maps onto a neighbourhood of in

It is sometimes common to equate the mapping and the curve . If the curve is as above but instead in the complex plane, we can instead make the following equivalent definition.

Definition 2   A curve is said to be an analytic curve (or analytic arc) if every point of has an open neighbourhood for which there is an onto conformal map (where is the unit disc) such that is mapped onto by

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 . 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.

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

Bibliography

1

Theodore B. Gamelin. Complex Analysis. Springer-Verlag, New York, New York, 2001.

2

Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.