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In this article, we presumed that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and creative approach, here we study k -almost Yamabe solitons and gradient k -almost Yamabe solitons. First, two examples are constructed to ensure the existence of gradient k -almost Yamabe solitons. Then we show that if a perfect fluid spacetime admits a k -almost Yamabe soliton, then its potential vector field is Killing if and only if the divergence of the potential vector field vanishes. Besides, we prove that if a perfect fluid spacetime permits a k -almost Yamabe soliton (g, k, ρ, λ), then the integral curves of the vector field ρ are geodesics, the spacetime becomes stationary and the isotopic pressure and energy density remain invariant under the velocity vector field ρ. Also, we establish that if the potential vector field is pointwise collinear with the velocity vector field and ρ(a) = 0 where a is a scalar, then either the perfect fluid spacetime represents a phantom era, or the potential function Φ is invariant under the velocity vector field ρ. Finally, we prove that if a perfect fluid spacetime permits a gradient k -almost Yamabe soliton (g, k,DΦ, λ) and R, λ, k are invariant under ρ, then the vorticity of the fluid vanishes. [ABSTRACT FROM AUTHOR]

In this paper, we classify 3-dimensional Riemannian manifolds endowed with a special type of vector field if the Riemannian metrices are Ricci-Yamabe solitons and gradient Ricci-Yamabe solitons, respectively. Finally, we construct an example to illustrate our result. [ABSTRACT FROM AUTHOR]

The object of the present paper is to obtain a necessary and sufficient condition for a 3-dimensional generalized (k, µ)-contact metric manifold to be locally φ-symmetric in the sense of Takahashi and the condition is verified by an example. Next we characterize a 3-dimensional generalized (k, µ)-contact metric manifold satisfying certain curvature conditions on the concircular curvature tensor. Finally, we construct an example of a generalized (k, µ)-contact metric manifold to verify Theorem 1 of our paper. [ABSTRACT FROM AUTHOR]