Reeb Flow Symmetry on Almost Cosymplectic Three-Manifolds

Abstract

Abstract. We prove that the Ricci operator S of an almost cosymplectic three-manifold M is invariant along the Reeb flow, that is, M satisfies £ ξ S = 0 if and only if M is either cosymplectic or locally isometric to the group E(1, 1) of rigid motions of Minkowski 2-space with a left invariant almost cosymplectic structure.

Remarks on Almost Cosymplectic 3-Manifolds with RICCI Operators

“…Because the Ricci operator is transversely Killing, from (12), we have (∇ e 1 Q)e 1 � 0 and (∇ e 2 Q)e 2 � 0, which are compared with (14) and (15), respectively, implying e 1 σ e 1 − 1 2λ σ e 2 e 2 (λ) + σ e 1 � 0,…”

“…In recent years, many classification results on almost cosymplectic manifolds of dimension three emerged. For example, Cho, in [15], studied Reeb flow symmetry (that is, the Ricci tensor is invariant along the Reeb flow) on almost cosymplectic 3-manifolds. Moreover, semisymmetry, local ϕ-symmetry, curvature, and ball homogeneities on almost cosymplectic 3-manifolds were considered in [3,4,8,9], respectively.…”

Trans-Sasakian 3-Manifolds with Reeb Flow Invariant Ricci Operator

“…The Reeb vector field is an eigenvector field of the Ricci operator. The above lemma can be seen in [19] Lemma 5. The Ricci operator on an α-Sasakian 3-manifold is invariant along the Reeb flow.…”

“…On a Riemannian manifold, we have divQ = 1 2 ∇r. In this context, it is equivalent to g((∇ ξ Q)ξ + (∇ e Q)e + (∇ φe Q)φe, X) = 1 2 X(r) (19) for any vector field X. Replacing X in (19) by ξ and recalling (16) and the first term of (12), we obtain 2β(A − 2α 2 + 2β 2 + 2ξ(β)) = 0, or equivalently,…”

“…Cho in [18] proved that a contact metric 3-manifold satisfies Equation (1) if and only if it is Sasakian or locally isometric to SU(2) (or SO(3)), SL(2, R) (or O(1, 2)), the group E(2) of rigid motions of Euclidean 2-plane. Cho in [19] proved that an almost cosymplectic 3-manifold satisfies (1) if and only if it is either cosymplectic or locally isometric to the group E(1, 1) of rigid motions of Minkowski 2-space. In addition, Cho and Kimura in [20] proved that an almost Kenmotsu 3-manifold satisfies (1) if and only if it is of constant sectional curvature −1 or a non-unimodular Lie group.…”