| course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
28 |
|
Paper 4, Section II, E |
2019 |
Let $H={x+i y \mid x, y \in \mathbb{R}, y>0}$ be the upper-half plane with hyperbolic metric $\frac{d x^{2}+d y^{2}}{y^{2}}$. Define the group $P S L(2, \mathbb{R})$, and show that it acts by isometries on $H$. [If you use a generation statement you must carefully state it.]
(a) Prove that $P S L(2, \mathbb{R})$ acts transitively on the collection of pairs $(l, P)$, where $l$ is a hyperbolic line in $H$ and $P \in l$.
(b) Let $l^{+} \subset H$ be the imaginary half-axis. Find the isometries of $H$ which fix $l^{+}$ pointwise. Hence or otherwise find all isometries of $H$.
(c) Describe without proof the collection of all hyperbolic lines which meet $l^{+}$with (signed) angle $\alpha, 0<\alpha<\pi$. Explain why there exists a hyperbolic triangle with angles $\alpha, \beta$ and $\gamma$ whenever $\alpha+\beta+\gamma<\pi$.
(d) Is this triangle unique up to isometry? Justify your answer. [You may use without proof the fact that Möbius maps preserve angles.]