Riemannian Submersion whose Total Manifold admits an Almost Hyperbolic Ricci Soliton

Abstract

In the present paper, we investigate geometric properties of a Riemannian submersion for which the total manifold is endowed with hyperbolic and almost hyperbolic Ricci solitons. We establish conditions under which the fibers of the submersion admit hyperbolic and almost hyperbolic Ricci solitons. Further, we derive necessary conditions for the base manifold to be an Einstein and a hyperbolic Ricci soliton. Also, we determine the scalar curvature associated with the fibers and the base manifold. Moreover, when the total space admits an almost hyperbolic Ricci soliton with a torse-forming potential vector field, we obtain conditions ensuring that the extrinsic vertical and horizontal distributions are almost generalized quasi-Einstein. We also obtain necessary conditions for the fibers and base manifold to be an Einstein and almost generalized quasi-Einstein manifold. Finally, examples are included to illustrate the existence of hyperbolic and almost hyperbolic Ricci solitons in Riemannian submersion.