Yamabe solitons on 3-dimensional cosymplectic manifolds

Abstract

In this paper, it has been proved that if a 3-dimensional cosymplectic manifold $M^3$ admits a Yamabe soliton, then either $M^3$ is locally flat or the potential field is a contact vector field. Some special potential vector fields of Yamabe solitons on 3-dimensional cosymplectic manifolds have been considered and some other results have been obtained. Also, for general $(2n+1)$-dimensional case, it will be shown that if an $f-$cosymplectic manifold $M^{2n+1}$ admits a contact Yamabe soliton structure, then $M^{2n+1}$ is a cosymplectic manifold. Finally, an example of Yamabe soliton on a 3-dimensional cosymplectic manifold is provided.

Keywords:

• contact geometry
• cosymplectic manifold
• yamabe soliton

PUBLIC INTEREST STATEMENT

The notion of Yamabe flow was introduced by R. S. Hamilton in 1988 in order to study Yamabe’s conjecture stating that any metric is conformally related to a metric with constant scalar curvature. In this research article, we discuss the geometry of Yamabe solitons on 3-dimensional cosymplectic manifolds. We prove that if a 3-dimensional cosymplectic manifold M³ admits a Yamabe soliton, then either M³ is locally flat or the potential field is a contact vector field. We consider some special potential vector fields of Yamabe solitons on 3-dimensional cosymplectic manifolds and obtain some other results. Also, for general (2n + 1)-dimensional case, we will show that if an f - cosymplectic manifold M^{2n + 1} admits a contact Yamabe soliton structure, then M^{2n + 1} is a cosymplectic manifold.

Acknowledgements

The authors would like to thank the anonymous referee(s) and editor for reading the manuscript carefully and giving valuable suggestions on it.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Hajar Ghahremani-Gol

The first author has received his Ph.D. degree in the joint supervision Dr Boroojerdian at Amirkabir University of Technology. Dr Fasihi is an Assistance professor of mathematics and faculty member at Imam Khomeini International University. He has published some research papers in widespread international journals. His interested are differential geometry, theoritical and Mathematical physics.

The second author has received her Ph.D. degree in the joint supervision Professor Razavi and Dr Didehvar at Amirkabir University of Technology. Dr Ghahremani-Gol currently works as an Assistant Professor at Shahed University. The research area of Ghahremani-Gol is Riemannian geometry, Mathematical fluid dynamics and she has published some research papers around them.