Study of gradient solitons in three dimensional Riemannian manifolds
Commun. Korean Math. Soc. Published online May 13, 2022
Uday Chand De and Gour Gopal Biswas
Department of Pure Mathematics, University of Calcutta; Department of Mathematics, University of Kalyani
Abstract : We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric P-connection. At first it is proven that if the metric of such a manifold is gradient m-quasi-Einstein metric, then either the gradient of the potential function ψ is collinear with the vector field P or, λ = -(m + 2) and the manifold is of constant sectional curvature -1, provided Pψ ≠ m. Next it is shown that if the metric of the manifold under consideration is gradient ρ-Einstein soliton, then the gradient of the potential function is collinear with the vector
field P. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric P-connection is gradient ω-Ricci soliton, then the manifold is of constant sectional curvature -1 and λ + μ = -2. Finally, we consider an example to verify our results.