A note on trans-Sasakian manifolds

Abstract

ABSTRACT. In this paper, we obtain some sufficient conditions for a 3-dimensional compact trans-Sasakian manifold of type (α, β) to be homothetic to a Sasakian manifold. A characterization of a 3-dimensional cosymplectic manifold is also obtained.

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

“…In this section, we give a proof of our main result Theorem 1. First, we introduce the following two important lemmas (see [12]) which are useful for our proof.…”

“…Let M be a trans-Sasakian 3-manifold and e be a unit vector field orthogonal to ξ. Then, {ξ, e, φe} forms a local orthonormal basis on the tangent space for each point of M. The Levi-Civita connection ∇ on M can be written as the following (see [12]) ∇ ξ ξ =0, ∇ ξ e = λφe, ∇ ξ φe = −λe, ∇ e ξ =βe − αφe, ∇ e e = −βξ + γφe, ∇ e φe = αξ − γe,…”

“…Under what condition is a trans-Sasakian 3-manifold proper? De [7][8][9][10][11][12], Deshmukh [13][14][15], Wang and Liu [16] and Wang [2,17] answered this question from various points of view. In this paper, we study this question under a new geometric condition.…”

Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds

“…In References [10][11][12][13][14][15][16][17][18][19], the authors studied compact TRS-manifolds with some restrictions on the smooth functions α, β and the vector field t appearing in their definition for getting conditions under which a TRS-manifold is homothetic to a Sasakian manifold. It is known that a compact simply connected TRS-manifold satisfying Poisson equations ∆α = β, ∆α = α 2 β, respectively, gives a necessary and sufficient condition for it to be homothetic to a Sasakian manifold (cf.…”

A remark on trans-Sasakian 3-manifolds

“…Therefore, to find on what condition a trans-Sasakian 3-manifold is proper is an interesting problem. Recently, S. Desmukh et al in [8,9,10,11,12] obtained various conditions under which a compact trans-Sasakian 3-manifold is homothetic to either a Sasakian or a cosymplectic 3-manifold. Trans-Sasakian 3-manifolds under some curvature restrictions were also studied by U. C. De et al in [5,6,7] and Wang [24].…”