Ghodratallah Fasihi-Ramandi

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 $g_{S_a}=\frac{{\pi }^2}{a^2{\sin }^2(\pi y/a)}(d{x}^2+d{y}^2)$, 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 $\mathfrak{sl}(2,\mathbb{R})$, 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 $\lambda \ne 0$, 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 $\mathfrak{sl}(2,\mathbb{R})\oplus \mathbb{R}$, 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 $\nu =2$.

This paper introduce the idea of second Ricci solitons. A second Ricci soliton is nothing but a steady hyperbolic Ricci soliton. We study the geometry of closed and compact second Ricci soliton manifolds. Immersed submanifolds as second solitons also will be investigated. Finally, we investigate this structure on warped product manifolds.