Abstract
PurposeThe authors set the goal to find the solution of the Eisenhart problem within the framework of three-dimensional trans-Sasakian manifolds. Also, they prove some results of the Ricci solitons, η-Ricci solitons and three-dimensional weakly symmetric trans-Sasakian manifolds. Finally, they give a nontrivial example of three-dimensional proper trans-Sasakian manifold.Design/methodology/approachThe authors have used the tensorial approach to achieve the goal.FindingsA second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.Originality/valueThe authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.
Highlights
Let the product manifold M~ 1⁄4 M 3R, known as an almost Hermitian manifold, possesses an almost complex structure J and the product metric G, where M is a ð2n þ 1Þ-dimensional almost contact metric manifold [1] and R, a real line
Many geometers characterized the geometrical properties of ð2n þ 1Þ-dimensional trans-Sasakian
The Eisenhart problems of finding the properties of second-order parallel tensors have been locally studied by Eisenhart and Levy, whereas Sharma [13] has solved the same problem globally on complex space form
Summary
Let the product manifold M~ 1⁄4 M 3R, known as an almost Hermitian manifold, possesses an almost complex structure J and the product metric G, where M is a ð2n þ 1Þ-dimensional almost contact metric manifold [1] and R, a real line. There was a natural question “Up to which dimension, the proper trans-Sasakian manifold exists”? Marrero [3], in 1992, gave an affirmative answer of this question He proved that a transSasakian manifold of dimension ≥5 is either a cosymplectic manifold, or an α-Sasakian manifold or a β-Kenmotsu manifold. In 1923, Eisenhart showed that if a positive definite Riemannian manifold admits a second-order parallel symmetric covariant tensor other than a constant multiple of the metric tensor, it is reducible [11]. The Eisenhart problems of finding the properties of second-order parallel tensors have been locally studied by Eisenhart and Levy, whereas Sharma [13] has solved the same problem globally on complex space form. For some deep results on this topic, we recommend [14,15,16,17,18] and the references therein
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