In this paper, we studied the geometrical aspects of a perfect fluid spacetime with torseforming vector field ξ under certain curvature restrictions, and Ricci-Yamabe soliton and η-Ricci-Yamabe soliton in a perfect fluid spacetime. Conditions for the Ricci-Yamabe soliton to be steady, expanding or shrinking are also given. Moreover, when the potential vector field ξ of η-Ricci-Yamabe soliton is of gradient type, we derive a Poisson equation and also looked at its particular cases. Lastly, a non-trivial example of perfect fluid spacetime admitting η-Ricci-Yamabe soliton is constructed.

The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere $$S^n(r)$$ S n ( r ) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with $$\alpha \ne 0$$ α ≠ 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space $$E^n$$ E n or Euclidean sphere $$S^n.$$ S n . Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.

In this article, we derive Chen’s inequalities involving Chen’s δ-invariant δM, Riemannian invariant δ(m1,⋯,mk), Ricci curvature, Riemannian invariant Θk(2≤k≤m), the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we first derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms.

In the present paper, we study the invariant submanifolds of [Formula: see text]-Kenmotsu manifolds. Firstly, we show that any invariant submanifold of [Formula: see text]-Kenmotsu manifold is again [Formula: see text]-Kenmotsu manifold and minimal. Then, we give some characterizations of totally geodesic submanifolds of [Formula: see text]-Kenmotsu manifolds. We study 3-dimensional submanifold and prove that a 3-dimensional submanifold of [Formula: see text]-Kenmotsu manifold is totally geodesic if and only if it is invariant. Also, [Formula: see text]-Ricci soliton is considered on invariant submanifold of [Formula: see text]-Kenmotsu manifolds. Lastly, the non-trivial examples are constructed to verify some of our obtained results.

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In this paper, we study the generalized m-quasi-Einstein metric in the context of contact geometry. First, we prove if an H-contact manifold admits a generalized m-quasi-Einstein metric with non-zero potential vector field V collinear with ?, then M is K-contact and ?-Einstein. Moreover, it is also true when H-contactness is replaced by completeness under certain conditions. Next, we prove that if a complete K-contact manifold admits a closed generalized m-quasi-Einstein metric whose potential vector field is contact thenMis compact, Einstein and Sasakian. Finally, we obtain some results on a 3-dimensional normal almost contact manifold admitting generalized m-quasi-Einstein metric.

This paper examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci–Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be [Formula: see text]-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes a Ricci–Yamabe soliton under certain restrictions. In this series, it is proven that a [Formula: see text]-dimensional [Formula: see text]-AKM equipped with a gradient ARYS is either locally isometric to [Formula: see text] or the Reeb vector field and the soliton vector field are codirectional. The properties of three-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied.