Let $\mathbb{H}$ be the Poincare half plane with the hyperbolic metric. Let $Gl(2,\mathbb{R})$ be equipped with a left invariant metric?



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> Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to $\mathbb{H}$? If yes, what is a precise formula for such a Riemannian submersion?
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Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to the Poincare half plane?
Define the hyperbolic metric $g$ on $\mathbb{H}^2$ by 
$$
g = \frac{1}{2y^2} (d x^2 + d y^2).
$$
The Iwasawa decomposition yields an isomorpism $SL(2)/SO(2)=\mathbb{H}^2$ which is given by
$$
\pmatrix{a\_{11} & a\_{12}\\ a\_{21} & a\_{22}} \mapsto \frac{1}{(a\_{21}^2 + a\_{22}^2)}(a\_{11} a\_{21} + a\_{12} a\_{22}, 1).
$$
Moreover, if you define a metric $h$ on $SL(2)$ by 
$$
h\_A (V, W) = Tr(A^{-1} V A^{-1} W),
$$
then the projection $SL(2) \to \mathbb{H}^2$ is a Riemannian submersion. Some time ago, I've written [some notes on this](https://share.cocalc.com/share/f26774ef-f5c4-4e05-9fd3-a235d176f341/SLmodSOtoPlaneAndSym.sagews?viewer=share) (and related things concerning the Poincare upper half plane). They are not in a very good shape but may be helpful nonetheless (they were meant as a first experiment with sage). 


