Journal of Mathematical Physics, Analysis, Geometry 2019, vol. 15, No 3, pp. 307-320   https://doi.org/10.15407/mag15.03.307     ( to contents , go back )

### Ricci Solitons and Gradient Ricci Solitons on N(k)-Paracontact Manifolds

Uday Chand De

Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kol-700019, West Bengal, India
E-mail: uc de@yahoo.com

Krishanu Mandal

Department of Mathematics, K.K. Das College, GRH-17, Baishnabghata-Patuli, Kol-700084, West Bengal, India
E-mail: krishanu.mandal013@gmail.com

Received February 14, 2018, revised June 1, 2018.

Abstract

An $\eta$-Einstein paracontact manifold $M$ admits a Ricci soliton $(g,\xi)$ if and only if $M$ is a $K$-paracontact Einstein manifold provided one of the associated scalars $\alpha$ or $\beta$ is constant. Also we prove the non-existence of Ricci soliton in an $N(k)$-paracontact metric manifold $M$ whose potential vector field is the Reeb vector field $\xi$. Moreover, if the metric $g$ of an $N(k)$-paracontact metric manifold $M^{2n+1}$ is a gradient Ricci soliton, then either the manifold is locally isometric to a product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of negative constant curvature equal to $-4$, or $M^{2n+1}$ is an Einstein manifold. Finally, an illustrative example is given.

Mathematics Subject Classification 2000: 53B30, 53C15, 53C25, 53C50, 53D10, 53D15.
Key words: paracontact manifold, N(k)-paracontact manifold, Ricci soliton, gradient Ricci soliton, Einstein manifold.