Classification of general eta-hyperbolic Ricci solitons on the Sasaki unit tangent bundle

Classify η-hyperbolic Ricci solitons with arbitrary smooth potential vector fields on the unit tangent bundle T^1S_a of the hyperbolic strip, beyond the trivial Killing-potential solutions and the explicitly analyzed potential classes.

Background

The paper introduces η-hyperbolic Ricci solitons on (T1S_a,gS) by adding a term ν η⊗η to the hyperbolic Ricci soliton equation. It establishes that, for Killing potentials and several restricted classes of θ-invariant potentials, the solutions are trivial: the potential is Killing, μ = -3/2, and ν = 2.

For general θ-invariant potentials, the authors derive a reduced system on the hyperbolic strip. The paper does not resolve whether nontrivial solutions exist for arbitrary potentials outside the classes treated, leaving the general classification unresolved.

References

Finally, the $\eta$-Ricci solitons of $gS$ possess a Killing potential with parameters $\lambda=\tfrac32$ and $\mu=-2$, while for $\eta$-hyperbolic Ricci solitons, we obtain only trivial solutions (a Killing potential with $\mu=-\tfrac32$, $\nu=2$, and $\lambda$ arbitrary) within the same potential classes.

— $η-$ Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip  (2610.11705 - Sy et al., 8 Oct 2026) in Section 1, Introduction