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 Sa={(x,y)∈ℝ2:0<y<a} endowed with the metric gSa=π2/a2sin2(πy/a)(dx2+dy2), and on its unit tangent bundle T1Sa endowed with the Sasaki metric gS. On the strip, we show that the Gauss curvature is constant equal to −1, that the Lie algebra of Killing fields is isomorphic to sl(2,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≠0, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of (T1Sa,gS) is four-dimensional and isomorphic to sl(2,ℝ)⊕ℝ, that gS is not Einstein but is η-Einstein with constant coefficients, RicS=−3/2 gS+2 η⊗η, and that (T1Sa,gS) 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 gS have a Killing potential and λ=3/2, 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=−3/2 and ν=2.
Related
- Properly embedded Legendrian surfaces in the tangent sphere bundle of hyperbolic space
- Deformations of the logarithmic tangent bundle of smooth projective surfaces
- Scalar curvature bounds and tangent cones for Kähler--Ricci shrinkers
- On the tangent flows of noncollapsed Ricci flow limit spaces
- Parametrization of ancient Ricci flows with smooth compact tangent shrinkers
BibTeX
@misc{sy2026hyperbolic,
title = {{$η-$ Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip}},
author = {Amadou Sy and Ameth Ndiaye and Ghodratallah Fasihi-Ramandi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11705},
eprint = {2610.11705},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11705},
}
RIS
TY - UNPB
TI - η− Hyperbolic Ricci Solitons on the Unit Tangent Bundle of the Hyperbolic Strip
AU - Amadou Sy
AU - Ameth Ndiaye
AU - Ghodratallah Fasihi-Ramandi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11705
UR - https://arxiv.org/abs/2610.11705
AB - We study Killing vector fields, Ricci solitons and hyperbolic Ricci solitons on the hyperbolic strip S_a={(x,y)∈ℝ^2:0<y<a} endowed with the metric g_S_a=π^2/a^2sin^2(πy/a)(dx^2+dy^2), and on its unit tangent bundle T^1S_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 sl(2,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≠0, is a Killing field. On the unit tangent bundle, we prove that the Lie algebra of Killing fields of (T^1S_a,g^S) is four-dimensional and isomorphic to sl(2,ℝ)⊕ℝ, that g^S is not Einstein but is η-Einstein with constant coefficients, Ric^S=−3/2 g^S+2 η⊗η, and that (T^1S_a,g^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^S have a Killing potential and λ=3/2, 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=−3/2 and ν=2.
ER -