Conformal map - Knowlpedia

Definition

Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. A conformal map in the convention used here is a smooth map $f:M\to N$ for which there is a smooth positive function $\lambda:M\to(0,\infty)$ satisfying

f^*h=\lambda^2g.

Equivalently, every tangent map $df_p$ multiplies all lengths by the same factor $\lambda(p)$ , and therefore preserves angles. Positive-definiteness forces $df_p$ to be injective, so such a map is a smooth immersion and $\dim M\leq\dim N$ .

Equal dimensions and isometries

When $\dim M=\dim N$ , a conformal map is a local diffeomorphism. If it is globally a diffeomorphism, it is a conformal diffeomorphism or conformal transformation. The special case $\lambda=1$ is a Riemannian isometric immersion.

Some authors use “conformal map” only for equal-dimensional local diffeomorphisms or homeomorphisms and call the general pullback-preserving map above a conformal immersion. The displayed metric equation is the house convention; it makes the dimension and rank requirements unambiguous.

Oriented surfaces and holomorphic maps

An orientation and Riemannian metric on a real surface determine the complex structure whose multiplication by $i$ rotates tangent vectors through $+\pi/2$ . Conversely, a Riemann surface determines an oriented conformal class of Riemannian metrics.

For maps between oriented Riemannian surfaces, a conformal local diffeomorphism is orientation-preserving exactly when it is holomorphic for the associated Riemann-surface structures. It is orientation-reversing exactly when it is antiholomorphic. Thus “conformal” alone does not encode orientation.

A nonconstant holomorphic map may have critical points. It is conformal in the strict positive-factor sense away from those points, but at a critical point its pullback metric has zero scale factor. Such a map is often called weakly conformal rather than conformal everywhere.

Classical complex-analysis examples

The Riemann mapping theorem produces a biholomorphic—and hence orientation-preserving conformal—diffeomorphism from every nonempty proper simply connected plane domain to the unit disc. Every Möbius transformation is an orientation-preserving conformal diffeomorphism of the Riemann sphere. In fact, the Möbius group is the full orientation-preserving conformal diffeomorphism group of the round sphere; adjoining anti-Möbius maps gives the orientation-reversing component.

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. DOI record. Relevant: Riemannian metrics, pullbacks, and conformal changes.
Otto Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer, 1981. DOI record. Relevant: §§1–2, holomorphic maps and conformal structures on surfaces.
Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapters 3–4, conformal maps, Möbius transformations, and the Riemann mapping theorem.