Let $(M,g)$ be a Riemannian manifold and $TM$ its tangent bundle, equipped with the Sasaki metric $g_S$. Let $T_1M$ be the unit tangent bundle (consisting of all tangent vectors of unit length), with metric given by restriction of $g_S$.
My question is: Does an isometry of $T_1M$ preserve its $S^1$-fibre structure? Or, phrased differently: Are the vertical and horizontal subbundle of $TT_1M$ (or even $TTM$) preserved under the differential of such an isometry? If so, is there any easy argument to show this? Or some reference?
Maybe some context: I am currently studying Thurston's eight geometries, in particular the Seifert fibred ones. One of them, the space $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ can be seen as the universal cover of $\mathrm{PSL}_2(\mathbb{R})$ (the projective linear group), which is isomorphic to the orientation preserving isometry group of $\mathbb{H}^2$. Since this groups acts transitively on $T_1\mathbb{H^2}$ with trivial stabilizers, it can be identified with $T_1\mathbb{H}^2$. Then, we take the Sasaki metric on $T_1\mathbb{H}^2$ and pull it back to get a metric on $\widetilde{\mathrm{SL}}_2(\mathbb{R})$. Note that $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ is fibred by lines (the preimages of the $S^1$-fibres of $T_1\mathbb{H}^2$). On important feature is that every closed 3-manifold $M$ modelled on $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ (that is, $M$ is a quotient $\widetilde{\mathrm{SL}}_2(\mathbb{R})/\Gamma$, where $\Gamma$ is a discrete subgroup of $\mathrm{Isom}(\mathbb{H}^2)$ acting freely and properly discontinuously on $\widetilde{\mathrm{SL}}_2(\mathbb{R})$) is Seifert fibred. The argument involves that any isometry of $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ leaves its line fibration invariant. This should be the same as saying that any isometry of $T_1\mathbb{H}^2$ leaves its $S^1$-fibration invariant (which, on a tangent level, corresponds to the vertical subbundle). See also Scott (1983)
Any help would be highly appreciated!