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

Abstract: We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip $S_a={(x,y)\in\mathbb{R}<sup>2:0&lt;y&lt;a}$ endowed with the metric $$g_{S_{a}}=\frac{π<sup>2}{a<sup>2\sin<sup>2(πy/a)}(dx<sup>2+dy<sup>2)$$, and on its unit tangent bundle $T<sup>1S_a$ endowed with the Sasaki metric $g<sup>S$. On the strip, we show that the Gauss curvature is constant equal to $-1$, that the Lie algebra of Killing fields is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, that every Ricci soliton is trivial (Killing potential, $λ=-1$), and that a $2$-Killing field which is conformal, or which is the potential of a hyperbolic Ricci soliton with $λ\neq0$, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of $(T<sup>1S_a,g<sup>S)$ is four-dimensional and isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$, that $g<sup>S$ is not Einstein but is $η$-Einstein with constant coefficients, $$\operatorname{Ric}<sup>S=-\tfrac32\,g<sup>S+2\,η\otimesη$$, and that $(T<sup>1S_a,g<sup>S)$ admits no Ricci soliton. For hyperbolic Ricci solitons we prove non-existence for Killing, left-invariant, and $θ$-invariant potentials with conformal projection, and we reduce the general $θ$-invariant case to a system of equations on the strip; the classification for an arbitrary potential is left open. Finally, we show that the $η$-Ricci solitons of $g<sup>S$ have a Killing potential and $λ=\tfrac32$, $μ=-2$, and that the $η$-hyperbolic Ricci solitons obtained in the same classes of potentials are the trivial ones, with a Killing potential, $μ=-\tfrac32$ and $ν=2$.