analytic curve (original) (raw)

Definition.

Suppose X is a real analytic manifold. A curve γ⊂X is an analytic curve if it is a real analytic submanifoldof dimension 1. Equivalently if near each point p∈γ,there exists a real analytic mappingf:(-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 equivalent definition.

Definition.

A curve γ⊂ℂ 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:𝔻→Δ (where 𝔻⊂ℂ is the unit disc) such that 𝔻∩ℝ is mapped onto Δ∩γ 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.

Note that locally all complex analytic subvarieties of dimension 1 in ℂ2 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.