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

Yan Zhao,Ximin Liu,Wenjie Wang

doi:10.3390/math6110246

Abstract

Let M be a three-dimensional trans-Sasakian manifold of type ( α , β ) . In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an α -Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Applying this, we give a new characterization of proper trans-Sasakian 3-manifolds.

Highlights

A trans-Sasakian manifold is usually denoted by ( M, φ, ξ, η, g, α, β), where both α and β are smooth functions and (φ, ξ, η, g) is an almost contact metric structure
When β = 0, α is a constant if dimM ≥ 5 and in this case M becomes an α-Sasakian manifold if α ∈ R∗ or a cosymplectic manifold if α = 0
Unlike the above case, when α = 0, β is not necessarily a constant even if dimM ≥ 5 or M is compact for dimension three

Summary

Introduction

A trans-Sasakian manifold is usually denoted by ( M, φ, ξ, η, g, α, β), where both α and β are smooth functions and (φ, ξ, η, g) is an almost contact metric structure. On an almost contact metric manifold ( M, φ, ξ, η, g), the Ricci operator of M is said to be Reeb flow invariant if it satisfies. 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. We obtain a new characterization of proper trans-Sasakian 3-manifolds by employing (1) and proving. The Ricci operator of a trans-Sasakian 3-manifold is invariant along Reeb flow if and only if the manifold is an α-Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Some corollaries induced from Theorem 1 are given in the last section

Trans-Sasakian Manifolds

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