Natural Metric on Tangent Bundle of Hyperbolic Plane?

Question

I am currently looking to understand the isometries of all the geometries described in W. Thurston's Geometrization Conjecture. I struggle with the details of how to construct the metric of Thurston's 6th geometry, that of $\widetilde{SL(2,\mathbb{R})}$. From the material that I've read (two sources below), the idea is to identify the unit tangent bundle of the Hyperbolic Plane, $UT(\mathbb{H}^2)$, with the Projective Special Linear Group $PSL(2,\mathbb{R})$ via fractional linear transformations and then pull-back the metric onto $\widetilde{SL(2,\mathbb{R})}$.

The construction usually starts with a presentation of the Hyperbolic Plane as the Poincaré upper half-plane $\mathbb{H^2}=\{z \in\mathbb{C}: \text{Im}(z)>0\}$, endowed with the metric $ds^2_{\mathbb{H}}= dzd\bar{z}/\text{Im}(z)^2$. Equivalently, for for $z =x + iy$, we may write $ds^2 = (dx^2+dy^2)/y^2$.

The Unit Tangent Bundle is then $UT(\mathbb{H^2}) = \{(z,\eta) \in T(\mathbb{H^2}) : \|\eta\|_\mathbb{H^2} = \|\eta\|_{Eucl.}/\text{Im}(z) = 1\}$, which is a circle bundle over $\mathbb{H^2}$.

All sources so far seem to agree that there is a natural metric on $UT(\mathbb{H^2})$ induced by the one on $\mathbb{H^2}$ without mentioning what this metric is. Further reading on natural metrics for Riemannian manifolds seems to suggest there are many natural metrics; but only one can induce the right geometry for $\widetilde{SL(2,\mathbb{R})}$. Clearly, I will find different isometries if I use \begin{equation} ds^2_{UT(\mathbb{H^2})} = (dx^2 + dy^2)/y^2 + d\phi^2, \end{equation} than if I use \begin{equation} ds^2_{UT(\mathbb{H^2})} = (dx^2 + dy^2 + d\phi^2)/y^2, \end{equation} where $\phi$ is the $S^1$ angle.

Can anyone here help me understand what is the correct one?

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Sources

The Eight Geometries of the Geometrization Conjecture (page 17)

The Geometry of 3-Manifolds (page 462, or page 62 of the document)

(I would post more, but Mathexchange won't let me post more than 2 links)

Answer 1

Although there is a choice in the construction of the metric on the Lie group $\Gamma = \widetilde{SL(2,\mathbb{R})}$, as there is for any Lie group $\Gamma$, the usual requirement is that the metric is invariant under the left-multiplication action $L_g : \Gamma \to \Gamma$ defined by $L_g(h)=gh$. This cuts down severely on the choices.

The usual convention for this choice is to pick an inner product on the tangent space of the identity element $T_e G$, and then transport that around to obtain an inner product on $T_g G$ for each $g \in G$ by requiring that $D L_g : T_e G \to T_g G$ preserve the inner product. This convention works just as well for $\widetilde{SL(2,\mathbb{R})}$ as for any other Lie group, and that's the main reason that you don't see very much in the way of a special discussion of the metric.

Now, perhaps you also want to have some kind of nice global formula. For this purpose I would recommend the discussion of the Liouville metric in Bonahon's paper "The geometry of Teichmüller space via geodesic currents"; the context there is not exactly the same as the context of $\widetilde{SL(2,\mathbb{R})}$, but it is very close and one should be able to make the transition between contexts without trouble.