Unit tangent sphere bundles with the Reeb flow invariant Ricci operator

Abstract

In this paper, we study unit tangent sphere bundles T₁M whose Ricci operator $\bar{S}$ is Reeb flow invariant, that is, L_ξ$\bar{S}$ = 0. We prove that for a 3-dimensional Riemannian manifold M, T₁M satisfies L_ξ$\bar{S}$ = 0 if and only if M is of constant curvature 1. Also, we prove that for a 4-dimensional Riemannian manifold M, T₁M satisfies L_ξ $\bar{S}$ = 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.