A remark on trans-Sasakian 3-manifolds

Abstract

Let M be a trans-Sasakian 3-manifold of type (α, β). In this paper, we give a negative answer to the question proposed by S. Deshmukh [Mediterr. J. Math. 13 (2016), no. 5, 2951-2958], namely we prove that the differential equation ∇β = ξ(β)ξ on M does not necessarily imply that M is homothetic to either a Sasakian or cosymplectic manifold even when M is compact. Many examples are constructed to illustrate this result.

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

“…Using this in (14), we have ∇β = ξ(β)ξ. The following proof follows directly from [2]. For sake of completeness, we present the detailed proof.…”

“…This conclusion is not necessarily true for dimension three. However, unlike the above case, when α = 0, β is not necessarily a constant even if dimM ≥ 5 or M is compact for dimension three (see [2]). The set of all trans-Sasakian manifolds of type (0, β) coincides with that of all f -cosymplectic manifolds (see [3]) or f -Kenmotsu manifolds (see [4][5][6]).…”

“…In view of this, we introduce an interesting question: Remark 3. Given a trans-Sasakian 3-manifold, following proof of Theorem 1, we still do not know whether β is a constant or not even when α = 0 and the manifold is compact (see [2]).…”

“…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.…”

Three-dimensional trans-Sasakian manifolds and solitons

“…Since then, the study of three-dimensional trans-Sasakian manifolds attract the researchers. For more details, we refer [4][5][6][7][8][9][10] and the references therein. The Eisenhart problem of finding the parallel tensors (symmetric and skew-symmetric) is an important subject in the differential geometry and its allied areas.…”