# Questions tagged [conformal-geometry]

A conformal structure is one that captures the idea of angles, but not lengths. Conformal geometry is the study of geometries with conformal structures, with special focus on transformations that preserves the angles (while possibly changing lengths). Examples of conformal transformations include the Mercator projection from cartography, and the Möbius transformations of the Riemann sphere.

1,180 questions
Filter by
Sorted by
Tagged with
56 views

### Is the complex tangent Injective?

So if we consider the Analytic function $f(z)=\tan(z)$ when $z\in \{z\in\mathbb{C}\text{ : } -\pi/2<Re(z)<\pi/2 \}$ is it injective ? Its derivative is non zero however this only guarantees ...
• 394
19 views

• 25
37 views

### Doubt in Mobius geometry

I am currently studying Mobius geometry. I found a group in Mobius geometry called Mobius group which contains Mobius transformations. I have the following doubt. Dose this group contain ...
• 719
39 views

### Area of image of holomorphic function on the disk [duplicate]

I'm working on an old qualifying exam problem, which asks me to show that if $f:\mathbb{D}\rightarrow \mathbb{C}$ is holomorphic and injective and $f'(0)=1$, then the area of $f(\mathbb{D})$ is at ...
1 vote
39 views

### Comparing areas between conformal metrics in $\mathbb{R}^2$

I would like to ask for a reference on the following subject: Let $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ be a radial positive function, i.e., $$f(x,y)=\lambda(x^2+y^2)$$ for some $C^\infty$ ...
• 2,243
1 vote
34 views

### Torus equation in conformal geometric algebra

How would you define a torus using conformal geometric algebra? Since CGA has circles as a primitive, It seems to me that we should be able to able to define a torus as a circle C rotated around a ...
• 111
15 views

1 vote
94 views

• 41
1 vote
82 views

### Prove that there is no conformal mapping between $\mathbb{D}\setminus\{0,1/2\}$ and $\mathbb{D}\setminus\{0,1/4\}$

I have to prove that there is no conformal mapping between $\mathbb{D}\setminus\{0,1/2\}$ and $\mathbb{D}\setminus\{0,1/4\}$. I honestly have no idea. Intuitively, I guess that the problem has to do ...
• 1,344
24 views

• 394
29 views

### Show that the conformally complete Schwarzschild spacetime is asymptotically flat at null infinity

I am trying to show that the conformal factor used to conformally complete the Schwarzschild spacetime renders it asymptotically flat at null infinity (according to the mathematical definition given ...
• 589
21 views

41 views

• 31
23 views

### Understanding the Harmonic Energy for Maps Between Riemann Surfaces？

I am trying to understanding this functional, the thing I got confused about is the meaning of term $u_z\overline{u}_{\overline{z}}+ \overline{u}_{z}u_{\overline{z}}$ ? Because harmonic map is almost ...
23 views

• 1,683
17 views

• 498
52 views

### Determining Conformal Map from Conformal Factor

I am working on a project and need to apply a certain (Lorentzian) conformal transformation, $\psi:\mathbb{R}^2\mapsto \mathbb{R}^2$, to a figure which I have generated numerically in Mathematica. I ...
1 vote