analytic curve

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


Suppose X is a real analytic manifold. A curve γX is an analytic curve if it is a real analytic submanifold of dimensionMathworldPlanetmath 1. Equivalently if near each point pγ, there exists a real analytic mapping f:(-1,1)X, such that f has nonvanishing differential and maps onto a neighbourhood of p in γ.

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


A curve γ is said to be an analytic curve (or analytic arc) if every point of γ has an open neighbourhood Δ for which there is an onto conformal map f:𝔻Δ (where 𝔻 is the unit discPlanetmathPlanetmath) such that 𝔻 is mapped onto Δγ 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.


An analytic curve is a complex analytic subvariety of dimension 1 of a complex manifoldMathworldPlanetmath.

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


  • 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