Timeline for Find the fixed geodesic of an orientation-preserving isometry of the $3D$ hyperboloid model

9 events

when toggle format	what		by	license	comment
S Dec 8, 2017 at 22:57	history	suggested	j0equ1nn	CC BY-SA 3.0	the real answer was in the comments do I summarized them in the answer
Dec 8, 2017 at 19:10	review	Suggested edits
S Dec 8, 2017 at 22:57
Dec 7, 2017 at 23:08	vote	accept	j0equ1nn
Dec 7, 2017 at 23:08	comment	added	j0equ1nn		Ah okay, I see what you mean. Thanks.
Dec 7, 2017 at 22:27	comment	added	Igor Rivin		It's in the link in my answer.
Dec 7, 2017 at 21:59	comment	added	j0equ1nn		That makes sense, but let's say I give you a specific matrix. Can we say exactly what the pair of light like vectors are? I have not found any literature on a conformal map from $\overline{\mathcal{H}^3}$ to $\mathcal{I}^3\cup\big(\big\{p\in\mathbb{R}^{3,1}\mid\mathrm{n}(p)=0\big\}/\sim\big)$.
Dec 7, 2017 at 21:34	comment	added	Igor Rivin		@j0equ1nn The two endpoints you mention in the OP give two (light-like) vectors. Their span is your geodesic.
Dec 7, 2017 at 20:39	comment	added	j0equ1nn		That works fine for points but is awkward to do on an entire geodesic in the upper half-plane. I'm looking for a way of characterizing the plane in $\mathbb{R}^4$ that intersects with $\mathcal{I}^3$ to give $g$. Since this depends only on the matrix entries, there should be some function $\mathrm{PSL}(2,\mathbb{C})\rightarrow\mathbb{R}^4\times\mathbb{R}^4$ that outputs this pair of vectors.
Dec 7, 2017 at 18:59	history	answered	Igor Rivin	CC BY-SA 3.0