Let $ M $ be the total space of a circle bundle $ S^1 \to M \to \Sigma_g $ for $ g \geq 2 $. Suppose that there exists a transitive action of $ \widetilde{SL_2(\mathbb{R})} $ on $ M $. Must $ M $ be diffeomorphic to the unit tangent bundle of $ \Sigma_g $?
Motivation:
$ \Sigma_g = SO_2(\mathbb{R})\setminus SL_2(\mathbb{R})/\pi_1(\Sigma_g) $ and the unit tangent bundle of $ \Sigma_g $ can be written $$ UT(\Sigma_g)\cong SL_2(\mathbb{R})/\pi_1(\Sigma_g) $$
For every closed, oriented circle bundle $S^1 \to M \to \Sigma_g$ over a closed, oriented surface of genus $g \ge 2$, if the Euler number $\chi \in \mathbb Z$ of the circle bundle is nonzero then $M$ is the quotient of $\widetilde{SL_2(\mathbb R)}$ modulo the left action of some lattice embedding $\pi_1 M \hookrightarrow \widetilde{SL_2(\mathbb R)}$. Therefore, the right action of $\widetilde{SL_2(\mathbb R)}$ on itself descends to a transitive right action on $M$. Since circle bundles $M$ exist over $S_g$ with every possible Euler number $\chi \in \mathbb Z$, and since the Euler number of the tangent circle bundle is equal to the Euler characteristic of $\chi(\Sigma_g)=2-2g$, that gives you many other examples of such transitive actions.