Why is it that linear fractional transformations cannot take the upper half-plane to the first quadrant? Problem: Is there a linear fractional transformation that takes the upper half-plane to the first quadrant?  If yes, write it down.  If no, give proof.


Attempt: 


We have that a linear fractional transformation $f$ is a transformation of form

$$
f(z) = {az + b \over cz + d} \text{ s.t. } ad - bc \ne 0
$$

and where $a,b,c,d \in \mathbb{C}$.
Note by another theorem, $f$ must send circles and lines to circles or lines.
I think (but am not certain) that linear fractional transformations must take boundaries to boundaries.
Hence if the supposition were true, we would have that the $x$-axis of $\mathbb{C}$ is sent to the boundary of the first quandrant (i.e., the union of the positive reals, zero, and the pure imaginary numbers from $0$ upwards).  This violates (2).
Then the argument is complete.


Is this correct?
No. Linear fractional transformations naturally act on the Riemann sphere ℂP1 by biholomorphisms. (In fact, all biholomorphisms of ℂP1 are linear fractional transformations, but we don’t need it.) A biholomorphism, i.e. a holomorphic map with holomorphic inverse, is also a diffeomorphism: any Riemannian surface is also a smooth 2-dimensional real manifold.

A half-plane on ℂ is a half of Riemann sphere bounded by a great circle. Its boundary is smooth. The boundary of a quadrant isn’t smooth.

This was a conceptual proof that used “higher mathematics”, but I’m sure one can also present an elementary proof.
No. Linear fractional transformations naturally act on the Riemann sphere ℂP1 by biholomorphisms. (In fact, all biholomorphisms of ℂP1 are linear fractional transformations, but we don’t need it.) A biholomorphism, i.e. a holomorphic map with holomorphic inverse, is also a diffeomorphism: any Riemannian surface is also a smooth 2-dimensional real manifold.

A half-plane on ℂ is a half of Riemann sphere bounded by a great circle. Its boundary is smooth. The boundary of a quadrant isn’t smooth.

This was a conceptual proof that used “higher mathematics”, but I’m sure one can also present an elementary proof.