Abstract
Purpose
The 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
Design/methodology/approach
The authors have used the tensorial approach to achieve the goal.
Findings
A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g.
Originality/value
The authors declare that the manuscript is original and it has not been submitted to any other journal for possible publication.
Keywords
Citation
Chaubey, S.K. and De, U.C. (2022), "Three-dimensional trans-Sasakian manifolds and solitons", Arab Journal of Mathematical Sciences, Vol. 28 No. 1, pp. 112-125. https://doi.org/10.1108/AJMS-12-2020-0127
Publisher
:Emerald Publishing Limited
Copyright © 2021, Sudhakar Kumar Chaubey and Uday Chand De
License
Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Let the product manifold
The Eisenhart problem of finding the parallel tensors (symmetric and skew-symmetric) is an important subject in the differential geometry and its allied areas. 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, then it is reducible [11]. In Ref. [12], Levy presented that a second-order parallel symmetric nondegenerate tensor in a space form is proportional to the metric tensor. 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. Since then, many geometers studied the Eisenhart problems on different geometrical structures. For some deep results on this topic, we recommend [14–18] and the references therein.
A Ricci flow:
introduced by Hamilton [19] on a Riemannian manifold M, is used to solve the celebrated Poincaré conjecture [20, 21] and differentiable structure theorem (extension of Hamilton’s sphere theorem) [22]. It has applications in the string theory, thermodynamics, general relativity, cosmology, quantum field theory, etc. Also, the uniformization theorem and geometrization conjecture can be solved with the help of Ricci flow. Here g and S are the Riemannian metric and the Ricci tensor of M, respectively. Let the Ricci flow be governed by a one parameter family of diffeomorphisms and scalings, then its solution is known as a Ricci soliton. A Ricci soliton
A Riemannian metric g of a Riemannian manifold of dimension n is said to be an η-Ricci soliton [23] if it satisfies the equation
The above deep studies motivate us to characterize the three-dimensional trans-Sasakian manifolds whose metrics are almost Ricci solitons and η-Ricci solitons. We also find the solutions of the Eisenhart problems and properties of three-dimensional weakly
2. Basic results of trans-Sasakian manifolds and some definitions
A triplet
The curvature tensor R of a three-dimensional trans-Sasakian manifold satisfies [29
Setting
Also, from equation (2.5) we have
Next, it can be easily verified that M satisfies
The notion of quasi-conformal curvature tensor on a Riemannian manifold was introduced by Yano and Sewaki [30]. The quasi conformal curvature tensor
In 2011, Mantica and Molinari [32] introduced the notion of generalized
A three-dimension trans-Sasakian manifold M is said to be weakly
A trans-Sasakian manifold M is said to be an η-Einstein manifold if its nonzero Ricci tensor S satisfies
3. Second-order parallel symmetric tensor on three-dimensional trans-Sasakian manifolds
Let δ be a symmetric tensor of type
Setting
Again changing
Replacing
In consequence of Eqns (2.2)–(2.4) and (3.2), the above equation assumes the form
Changing
We suppose that α and β are nonzero and
This fact together with equation (3.2) reflect that
A second-order parallel symmetric tensor on a three-dimensional trans-Sasakian manifold is a constant multiple of the associated Riemannian metric g. In other words, the almost contact metric on a regular three-dimensional trans-Sasakian manifold is irreducible.
Next, we consider that the regular three-dimensional trans-Sasakian manifold is Ricci symmetric, that is,
This shows that the three-dimensional Ricci symmetric trans-Sasakian manifold is an Einstein manifold, provided
Every three-dimensional regular Ricci symmetric trans-Sasakian manifold M is an Einstein manifold, provided the ξ-sectional curvature tensor of M is non-zero.
Particularly, if
Every three-dimensional Ricci symmetric Sasakian (or Kenmotsu) manifold is Einstein.
For a three-dimensional Kenmotsu manifold, Corollary 3.3 has been proved by De and Pathak [41].
If possible, we suppose that the three-dimensional trans-Sasakian manifold
This equation together with equation (2.10) reflect that
Every three-dimensional Ricci symmetric trans-Sasakian manifold satisfies
If we take
A three-dimensional Ricci symmetric β-Kenmotsu (or α-Sasakian) manifold holds the relation
Let the three-dimensional trans-Sasakian manifold be Ricci symmetric, then it satisfies equation (3.8). In consequence of equations (2.5) and (3.8), we have
In consequence of equations (3.9) and (3.10), we find that
This shows that the trans-Sasakian manifold M of dimension three possesses a space of constant scalar curvature
Next, we suppose that M has a space of constant scalar curvature
The covariant derivative of (3.7) gives
A three-dimensional trans-Sasakian manifold M is Ricci symmetric if and only if it is a space of constant scalar curvature
Again from equation (3.11), we can state the following:
A three-dimensional Ricci symmetric Kenmotsu manifold is locally isometric to the hyperbolic space
A Ricci symmetric Sasakian manifold of dimension three is locally isometric to the sphere
In the light of equations (2.11), (3.7) and (3)–(3.12), we conclude that
A three-dimensional Ricci symmetric trans-Sasakian manifold M is quasi-conformally flat, but the converse is not true.
We suppose that a three-dimensional trans-Sasakian manifold is quasi-conformally flat, that is,
With the help of equations (2.6) and (3.13), we find
Since
By considering equations (2.3), (2.4),
Let M be a three-dimensional quasi-conformally flat trans-Sasakian manifold, then either M
is conformally flat or
possesses an almost η-Ricci soliton
if and only if .
In consequence of Theorem 3.10, we can state the following corollary.
A three-dimensional quasi-conformally flat trans-Sasakian manifold M with
Again, in view of equation (3.16) and Theorem 3.10, we can state the following:
The metric of a three-dimensional quasi-conformally flat β-Kenmotsu manifold with
A three-dimensional quasi-conformally flat α-Sasakian manifold with
A three-dimensional quasi-conformally flat Sasakian manifold satisfying
Every quasi-conformally flat Kenmotsu manifold of dimension three with
In view of equation (3.14), we can state:
Suppose a three-dimensional quasi-conformally flat trans-Sasakian manifold M satisfies
From equation (3.16), we have
Let a three-dimensional quasi-conformally flat trans-Sasakian manifold M satisfies
With the help of Theorem 3.17, we can state the following:
If a three-dimensional quasi-conformally flat β-Kenmotsu manifold M satisfies
Suppose a three-dimensional quasi-conformally flat α-Sasakian manifold M satisfies
Let us suppose that
Let
It is well-known that a three-dimensional trans-Sasakian manifold of type
If
A Ricci soliton
Suppose
Now, we define the following definitions as:
A vector field
A vector field
From the Definition 3.24 and Definition 3.25, it is clear that if a vector field is Killing then it is affine Killing, but converse is not, in general, true. Here we prove that the converse is true in a three-dimensional trans-Sasakian manifold, provided
Let us suppose that
An affine Killing vector field on a three-dimensional Kenmotsu (or Sasakian) manifold is Killing.
It is obvious that the metric tensor g is covariantly constant, that is,
A Ricci soliton
4. Three-dimensional weakly symmetric trans-Sasakian manifolds and Ricci flow
This section is dedicated to study the properties of three-dimensional weakly
Substituting
Again, replacing
By considering the above discussions, we can state the following:
The Ricci tensor of a three-dimensional weakly
If possible, we suppose that
A three-dimensional weakly
We know that a weakly
Every weakly Ricci symmetric trans-Sasakian manifold of dimension three with
In view of equations (3.15) and (4.5), we obtain
The metric of a three-dimensional weakly
Again from equations (3.15) and (4.6), we have
This reflects that a three-dimension weakly Ricci symmetric trans-Sasakian manifold with
A three-dimensional weakly Ricci symmetric trans-Sasakian manifold with
5. Example of proper three-dimensional trans-Sasakian manifold
Let
The Koszul's formula along with above results give
By the straightforward calculations, we can write
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Acknowledgements
The authors express their sincere thanks to the referee for his valuable comments in the improvement of the paper. The first author acknowledges authority of University of Technology and Applied Sciences-Shinas for their continuous support and encouragement to carry out this research work.