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

Abstract

In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold.

Preface to: Differential Geometry: Structures on Manifolds and Their Applications

“…One can distinguish several topics that have been investigated in the papers of this Special Issue. In addition to the proposed keywords we have already presented in the beginning, we also have some specific ones that appeared in the papers [1][2][3][4][5][6][7][8][9][10][11] that we emphasize in Table 2.…”

On Characterizing a Three-Dimensional Sphere

“…and D is the Levi-Civita connection with respect to the metric g (cf. [10][11][12][13][15][16][17]22]). We see that Equations ( 2) and (3) imply…”

“…A basic problem in studying the geometry of trans-Sasakian spaces consists in finding conditions under which such a space is homothetic to a Sasakian manifold (cf. [6,[9][10][11][12][13][14][15][16][17]). Naturally, a 3-dimensional sphere S 3 (c) of constant curvature c is a TRSM (S 3 (c), Ψ, ζ, η, g, α, β) with α = √ c and β = 0 (see next section).…”

Some New Results on Trans‐Sasakian Manifolds

“…However, such a property holds not necessarily true for general trans-Sasakian manifolds of dimension three. In the past decade, to determine on what geometric conditions a connected, compact, or complete trans-Sasakian three-manifold is proper has been proposed by Deshmukh in [8] and later considered by many authors (see recent results by De et al [9][10][11][12], Deshmukh et al [8,[13][14][15][16][17][18][19], Wang and Wang and Liu [20,21], Wang [4,22,23], Zhao [5,6] and Ma and Pei [24].…”