Weak Nearly Kenmotsu Manifolds and Gradient Ricci Solitons

Weak almost contact metric structures were recently introduced by Rovenski and Wolak (Indag Math (NS) 33(3):518–532, 2022). In this article, we introduce the concept of weak nearly Kenmotsu manifolds, which are an extension of the classical nearly Kenmotsu manifolds. We find a necessary and sufficient condition under which a weak nearly Kenmotsu manifold is nearly Kenmotsu. As a consequence of this result, we derive several curvature properties and extend some fundamental results of nearly Kenmotsu manifolds to weak nearly Kenmotsu manifolds, thereby generalizing the findings of Küpeli Erken et al. (Mediterr J Math 13(6):4497–4507, 2016). We also establish a necessary condition under which a hypersurface of a weak nearly Kähler manifold is weak nearly Kenmotsu. Finally, we present some results on weak nearly Kenmotsu manifolds equipped with a gradient Ricci soliton structure, which generalize the results of Ayar and Yıldırım (Facta Univ Ser Math Inform 34(3):503–510, 2019).