Timeline for Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to the Poincare half plane?

8 events

when toggle format	what		by	license	comment
Jan 17, 2019 at 5:19	vote	accept	Ali Taghavi
Dec 21, 2018 at 16:15	answer	added	Tobias Diez		timeline score: 6
Dec 21, 2018 at 6:40	comment	added	Thomas Richard		(The first arrow should be $M\mapsto \frac{1}{\|\det M\|^{1/2}}M$ and it works only on $GL(2,\mathbb{R})^+$.)
Dec 20, 2018 at 10:14	comment	added	Thomas Richard		The first map is just the the map $M\mapsto \tfrac{1}{\|\det(M)\|}M$. Maybe the map $f:SL(2,\mathbb{R})\to\mathbb{H}^2$ is easier to see, consider the action $(A,p)\mapsto A\cdot p$ of $SL(2,\mathbb{R})$ by isometries on $\mathbb{H}^2$. $f$ is then $A\mapsto A\cdot p_0$ where $p_0$ is some reference point. The isotropy group of $p_0$ is then some copy of $SO(2,\mathbb{R})$ inside $SL(2,\mathbb{R})$.
Dec 20, 2018 at 7:08	comment	added	Ali Taghavi		@Thomas Thank you for your comment. What do you mean by the quotiont in your first comment?Further more what is the map in the first arrows, is it the Grahm Schmitd map? If yes is not the range (SO2)?
Dec 19, 2018 at 10:07	comment	added	Thomas Richard		This at least gives a submersion, the remaining question is wether it is a Riemannien submersion for some metric of $GL(2,\mathbb{R})$.
Dec 19, 2018 at 8:19	comment	added	Thomas Richard		Have you tried considering the natural maps $GL(2,\mathbb{R})\to SL(2,\mathbb{R})\to SL(2,\mathbb{R})/SO(2,\mathbb{R})\sim \mathbb{H}^2$ and see if the metric on $\mathbb{H}^2$ comes from something interesting ?
Dec 19, 2018 at 5:58	history	asked	Ali Taghavi	CC BY-SA 4.0