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.
Highlights
It is well known that the product M = M × R of a (2n + 1)-dimensional almost contact metric manifold ( M, F, t, u, g)
Reference [4]), and the structure in the class W4 on ( M, J, g) gives a structure ( F, t, u, g, α, β) on M known as transSasakian structure
The class W4 should not be confused with Stiefel–Whitney characteristic class, but it is one of the sixteen classes specified by different combinations of covariant derivatives of the almost complex structure J on the almost Hermitian manifold
Summary
T satisfying T (t) = τ3 t, and we give necessary and sufficient conditions for M to be homothetic to a compact and connected Sasakian manifold (see Theorem 2). We show that a compact and connected TRS-manifold ( M, F, t, u, g, α, β) with Ricci curvature S(t, t) a non-zero constant and satisfying S(t, t) ≤ 2 α2 + β2 give necessary and sufficient conditions for M to be homothetic to a compact and connected Sasakian manifold (see Theorem 3). In the third result, we show that conditions S(t, t) 6= 0 and F ( gradα) = gradβ on a compact and connected TRS-manifold ( M, F, t, u, g, α, β) are necessary and sufficient for M to be homothetic to a compact and connected Sasakian manifold (see Theorem 4).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
