Conformal and non-conformal mappings

[JulieK]

My understanding is that conformal mapping is restricted to analytic functions.
What sort of mapping (if any) that can be used for non-analytic functions?

 

[andrewkirk]

My understanding is that conformal mapping is restricted to analytic functions.
What sort of mapping (if any) that can be used for non-analytic functions?

There are two non-equivalent definitions of conformal mapping.

One is as a mapping that locally preserves angles. The other is as a complex-valued function on an open set in $\mathbb{C}^n$ that is one-to-one and holomorphic.

Which one are you asking about?

 

[JulieK]

I am not sure they are not equivalent.
However, I am mainly interested in the second.

 

[lavinia]

My understanding is that conformal mapping is restricted to analytic functions.
What sort of mapping (if any) that can be used for non-analytic functions?

Conformal generally means infinitesimally angle preserving. Analytic functions are conformal away from singularities.

Analytic functions are also orientation preserving so an angle preserving map of the plane that reverses orientation will not be analytic.

On Riemannian manifolds a map f:M -> N is conformal if <df(x),df(y)> = g<x,y> where g is a positive function. This condition just says that the map is infinitesimally angle preserving. If f is a diffeomorphism then the pull back metric is said to be conformally equivalent to the original. In general metrics may not be conformally equivalent and each equivalence class is called a conformal structure.

A related idea is that of local conformal equivalence. For instance, a Riemannian manifold is said to be locally conformally flat if around each point there is an open neighborhood where the metric is conformally equivalent to the flat metric. This is true of all 2 dimensional Riemannian manifolds.

A conformal mapping does not have to be between manifolds of the same dimension. The condition, <df(x),df(y)> = g<x,y>, makes sense when M has lower dimension than N. For instance one may ask when a Riemannian manifold can be conformally immersed into another Riemannian manifold.