$\eta -$ Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip

We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip $S_a=\{(x,y)\in {\mathbb{R}}^2:0<y<a\}$ endowed with the metric $g_{S_a}=\frac{{\pi }^2}{a^2{\sin }^2(\pi y\mathrm{/}a)}(d{x}^2+d{y}^2)$, and on its unit tangent bundle $T^1{S}_a$ endowed with the Sasaki metric $g^S$. On the strip, we show that the Gauss curvature is constant equal to $-1$, that the Lie algebra of Killing fields is isomorphic to $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$, that every Ricci soliton is trivial (Killing potential, $\lambda =-1$), and that a $2$-Killing field which is conformal, or which is the potential of a hyperbolic Ricci soliton with $\lambda \mathrm{\ne }0$, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of $(T^1{S}_a,g^S)$ is four-dimensional and isomorphic to $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})\oplus \mathbb{R}$, that $g^S$ is not Einstein but is $\eta$-Einstein with constant coefficients, ${\mathrm{Ric}}^S=-\frac{3}{2}{g}^S+2\eta \otimes \eta$, and that $(T^1{S}_a,g^S)$ admits no Ricci soliton. For hyperbolic Ricci solitons we prove non-existence for Killing, left-invariant, and $\theta$-invariant potentials with conformal projection, and we reduce the general $\theta$-invariant case to a system of equations on the strip; the classification for an arbitrary potential is left open. Finally, we show that the $\eta$-Ricci solitons of $g^S$ have a Killing potential and $\lambda =\frac{3}{2}$, $\mu =-2$, and that the $\eta$-hyperbolic Ricci solitons obtained in the same classes of potentials are the trivial ones, with a Killing potential, $\mu =-\frac{3}{2}$ and $\nu =2$.