In this paper, we study unit tangent sphere bundles
T1M whose Ricci operator $\bar{S}$ is Reeb flow invariant, that is,
Lξ$\bar{S}$ = 0. We prove that for a 3-dimensional Riemannian manifold
M,
T1M satisfies
Lξ$\bar{S}$ = 0 if and only if
M is of constant curvature 1. Also, we prove that for a 4-dimensional Riemannian manifold
M,
T1M satisfies
Lξ $\bar{S}$ = 0 and ℓ$\bar{S}$ξ = 0 if and only if
M is of constant curvature 1 or 2, where ℓ = $\bar{R}$(·,ξ)ξ is the characteristic Jacobi operator.
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