How can I show that one can interchange any given points $p$ and $q$ on hyperboloid model(including two sheets) by an element of $O(n,1)$, Lorentz group?
Interchange points by Lorentz transformation
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riemannian-geometry
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0Sorry, I didn't fully register you said "interchange" in my first answer. – 2017-02-17
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So: given distinct points $p$ and $q$ on the quasisphere $ \{(x,y_1,\cdots,y_n):x^2-y_1^2-\cdots-y_n^2=1\} $, we want an element $g\in \mathrm{O}(1,n)$ such that $gp=q$ and $gq=p$, i.e. $g$ interchanges $p$ and $q$.
If $p$ and $q$ are on the same sheet, we can move $p$ to $n=(1,0,\cdots,0)$ with a 2D hyperbolic transformation $a$ on $\mathrm{span}(p,n)$ (assuming $p\ne n$) extended to $\mathbb{R}^{1,n}$, then apply another 2D hyperbolic rotation within $\mathrm{span}(ap,aq)$ so that they are symmetric across the $x$-axis, then apply a Euclidean rotation around the $x$-axis to swap them, then move them back to their original positions but swapped.
If $p$ and $q$ are on different sheets, we can apply a 2D hyperbolic rotation on them within their span until they are antipodal, in which case $-I$ will switch them before we put them back.