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In this research note, we investigated the characteristics of perfect fluid spacetime when coupled with the hyperbolic Ricci soliton. We additionally interacted with the perfect fluid spacetime, with a $ \varphi(\mathcal{Q}) $-vector field and a bi-conformal vector field that admits the hyperbolic Ricci solitons. Furthermore, we analyze the gradient hyperbolic Ricci soliton in perfect fluid spacetime, employing a scalar concircular field, and discuss about the gradient hyperbolic Ricci soliton's rate of change. In the end, we determined the energy conditions for perfect fluid spacetime in terms of gradient hyperbolic Ricci soliton with a scalar concircular field.

In this article, we delve into the study of statistical solitons on submanifolds of Kenmotsu statistical manifolds, introducing the presence of concircular vector fields. This investigation is further extended to study the behavior of almost quasi-Yamabe solitons on submanifolds with both concircular and concurrent vector fields. Concluding our research, we offer a compelling example featuring a 5-dimensional Kenmotsu statistical manifold that accommodates both a statistical soliton and an almost quasi-Yamabe soliton. This example serves to reinforce and validate the principles discussed throughout our study.

A solution to an evolution equation that evolves along symmetries of the equation is called a self-similar solution or soliton. In this manuscript, we present a study of η-Ricci solitons (η-RS) for an interesting manifold called the (ε)-Kenmotsu manifold ((ε)-KM), endowed with a semi-symmetric metric connection (briefly, a SSM-connection). We discuss Ricci and η-Ricci solitons with a SSM-connection satisfying certain curvature restrictions. In addition, we consider the characteristics of the gradient η-Ricci solitons (a special case of η-Ricci soliton), with a Poisson equation on the same ambient manifold for a SSM-connection. In addition, we derive an inequality for the lower bound of gradient η-Ricci solitons for (ε)-Kenmotsu manifold, with a semi-symmetric metric connection. Finally, we explore a number theoretic approach in the form of Pontrygin numbers to the (ε)-Kenmotsu manifold equipped with a semi-symmetric metric connection.

In this article, we investigate anti-invariant Riemannian and Lagrangian submersions onto Riemannian manifolds from the Lorentzian para-Sasakian manifold. We demonstrate that, for these submersions, horizontal distributions are not integrable and their fibers are not totally geodesic. As a result, they are not totally geodesic maps. The harmonicity of such submersions is also examined. We specifically prove that they are not harmonic when the Reeb vector field is horizontal. Finally, we provide an illustration of our findings and mention some number-theoretic applications for the same submersions.

The goal of the present research paper is to study how a spacetime manifold evolves when thermal flux, thermal energy density and thermal stress are involved; such spacetime is called a thermodynamical fluid spacetime (TFS). We deal with some geometrical characteristics of TFS and obtain the value of cosmological constant Λ. The next step is to demonstrate that a relativistic TFS is a generalized Ricci recurrent TFS. Moreover, we use TFS with thermodynamic matter tensors of Codazzi type and Ricci cyclic type. In addition, we discover the solitonic significance of TFS in terms of the Ricci metric (i.e., Ricci soliton RS).

This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal η-Ricci soliton and gradient conformal η-Ricci soliton with a potential vector field ζ. Additionally, we estimate the various conditions for which the target manifold of Riemannian submersion under the canonical variation is a conformal η-Ricci soliton with a Killing vector field and a φ(Ric)-vector field. Moreover, we deduce the generalized Liouville equation for Riemannian submersion under the canonical variation satisfying by a last multiplier Ψ of the vertical potential vector field ζ and show that the base manifold of Riemanian submersion under canonical variation is an η Einstein for gradient conformal η-Ricci soliton with a scalar concircular field γ on base manifold. Finally, we illustrate an example of Riemannian submersions between Riemannian manifolds, which verify our results.

In this paper, we first define a Kenmotsu-like statistical manifold (K.l.s.m) with examples. Then, we switch to Kenmotsu-like statistical submersions (K.l.s.s), where we investigate the fact that, for such submersions, each fiber is a statistical manifold that is similar to K.l.s.m, and the base manifold is similar to the Kähler-like statistical manifold. Subsequently, assuming the postulate that the curvature tensor with regard to the affine connections of the total space obeys certain criteria, we analyze such statistical submersions to those developed by Kenmotsu. Lastly, we talk about statistical submersions (SS) with conformal fibers (CFs) that are K.l.s.m.

We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson–Walker (GRW) spacetime, which is an eigenvector of the de Rham–Laplace operator. This paves the way to showing that the presence of a time-like special torse-forming vector field ξ with potential function ρ on a Lorentzian manifold (M,g), dimM>5, which is an eigenvector of the de Rham Laplace operator, gives a characterization of a GRW-spacetime. We show that if, in addition, the function ξ(ρ) is nowhere zero, then the fibers of the GRW-spacetime are compact. Finally, we show that on a simply connected Lorentzian manifold (M,g) that admits a time-like special torse-forming vector field ξ, there is a function f called the associated function of ξ. It is shown that if a connected Lorentzian manifold (M,g), dimM>4, admits a time-like special torse-forming vector field ξ with associated function f nowhere zero and satisfies the Fischer–Marsden equation, then (M,g) is a quasi-Einstein manifold.

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

f(R,T)-gravity is a generalization of Einstein’s field equations (EFEs) and f(R)-gravity. In this research article, we demonstrate the virtues of the f(R,T)-gravity model with Einstein solitons (ES) and gradient Einstein solitons (GES). We acquire the equation of state of f(R,T)-gravity, provided the matter of f(R,T)-gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a f(R,T)-gravity filled with perfect fluid admits an Einstein soliton (g,ρ,λ) and the Einstein soliton vector field ρ of (g,ρ,λ) is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the f(R,T)-gravity model. Next, we prove that if a f(R,T)-gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the f(R,T)-gravity model together with gradient Einstein soliton.

Purpose – The central idea of this research article is to examine the characteristics of Clairaut submersions from Lorentzian trans-Sasakian manifolds of type (α, β) and also, to enhance this geometrical analysis with some specific cases, namely Clairaut submersion from Lorentzian α-Sasakian manifold, Lorentzian β-Kenmotsu manifold and Lorentzian cosymplectic manifold. Furthermore, the authors discuss some results about Clairaut Lagrangian submersions whose total space is a Lorentzian trans-Sasakian manifolds of type (α, β). Finally, the authors furnished some examples based on this study. Design/methodology/approach – This research discourse based on classifications of submersion, mainly Clairaut submersions, whose total manifolds is Lorentzian trans-Sasakian manifolds and its all classes like Lorentzian Sasakian, Lorenztian Kenmotsu and Lorentzian cosymplectic manifolds. In addition, the authors have explored some axioms of Clairaut Lorentzian submersions and illustrates our findings with some non-trivial examples. Findings – The major finding of this study is to exhibit a necessary and sufficient condition for a submersions to be a Clairaut submersions and also find a condition for Clairaut Lagrangian submersions from Lorentzian trans-Sasakian manifolds. Originality/value – The results and examples of the present manuscript are original. In addition, more general results with fair value and supportive examples are provided.

In this research paper, we discussed some geometric axioms of a relativistic string cloud spacetime attached with strange quark matter. We determined the conformal $ \eta $-Ricci soliton on a relativistic string cloud spacetime attached with strange quark matter with a $ \varphi(\mathcal{R}ic) $-vector field. In addition, we illustrated some physical significance of conformal pressure $ P $ in terms of conformal $ \eta $-Ricci soliton with the same vector field. Besides this, we deduced a generalized Liouville equation from the conformal $ \eta $-Ricci soliton. Furthermore, we examine the harmonic relevance of conformal $ \eta $-Ricci soliton on string cloud spacetime attached with strange quark matter with a potential function $ \psi $. Finally, we turned up a necessary and sufficient condition for the 1-form $ \eta $, which is the $ {g} $-dual of the vector field $ \gamma $ on a string cloud spacetime attached with strange quark matter to be a solution for the Schrödinger-Ricci equation.

In this article, we investigate the Kenmotsu manifold when applied to a \(D_{\alpha}\)-homothetic deformation. Then, given a submanifold in a \(D_{\alpha}\)-homothetically deformed Kenmotsu manifold, we derive the generalized Wintgen inequality. Additionally, we find this inequality for submanifolds such as slant, invariant, and anti-invariant in the same ambient space.

In this paper, we develop the concept of gradient r-Almost Newton-Ricci-Yamabe solitons (in brief, gradient r-ANRY solitons) immersed in a Riemannian manifold. We deduce the minimal and totally geodesic criteria for the hypersurface of a Riemannian manifold in terms of the gradient r-ANRY soliton. We also exhibit a Schur-type inequality and discuss the triviality of the gradient r-ANRY soliton in the case of a compact manifold. Finally, we demonstrate the completeness and noncompactness of the r-Newton-Ricci-Yamabe soliton on the hypersurface of the Riemannian manifold.

In the present research note, we discuss the energy–momentum squared gravity model F(R,T2) coupled with perfect fluid. We obtain the equation of state for the perfect fluid in the F(R,T2)-gravity model. Furthermore, we deal with the energy–momentum squared gravity model F(R,T2) coupled with perfect fluid, which admits the hyperbolic Ricci solitons with a conformal vector field. We provide a clue in this series to determine the density and pressure in the radiation and phantom barrier periods, respectively. Also, we investigate the rate of change in hyperbolic Ricci solitons within the same vector field. In addition, we determine the different energy conditions, black holes and singularity conditions for perfect fluid attached to F(R,T2)-gravity in terms of hyperbolic Ricci solitons. Lastly, we deduce the Schrödinger equation for the potential Un with hyperbolic Ricci solitons in the F(R,T2)-gravity model coupled with perfect fluid and a phantom barrier.

The purpose of the article is to analyze the behavior of spacetime using a string cloud energy-momentum tensor $ \mathcal{T} $ having string cloud fluid density $ \rho $ and string tension $ \lambda $, named relativistic string cloud spacetime. We obtain some results for string cloud spacetime with a divergence-free matter tensor and a diminishing space matter tensor. Next, we discuss some curvature characteristics, such as conformally flat, Ricci semi-symmetric and pseudo-Ricci-symmetric, for relativistic string cloud spacetime. In addition, we gain a condition that coincides with the equation of state for the cloud of geometric strings in Ricci semi-symmetric string cloud spacetime.

In this article, we study the Ricci soliton on slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Moreover, we derive a lower-bound-type inequality for the slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection in terms of gradient Ricci solitons. We also characterize anti-invariant, invariant, quasi-umbilical submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection for which the same inequality case holds. Finally, we deduce the above inequalities in terms of a scalar concircular field on submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection.

In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or (α,β)-type almost contact manifolds endowed with the Schouten–Van Kampen connection (SVK-connection), including generalized normalized δ-Casorati Curvatures (δ-CC). We also discuss submanifolds on which the equality situations occur. Lastly, we provided an example derived from this research.

In this manifestation, we explain the geometrisation of η-Ricci–Yamabe soliton and gradient η-Ricci–Yamabe soliton on Riemannian submersions with the canonical variation. Also, we prove any fiber of the same submersion with the canonical variation (in short CV) is an η-Ricci–Yamabe soliton, which is called the solitonic fiber. Also, under the same setting, we inspect the η-Ricci–Yamabe soliton in Riemannian submersions with a φ(Q)-vector field. Moreover, we provide an example of Riemannian submersions, which illustrates our findings. Finally, we explore some applications of Riemannian submersion along with cohomology, Betti number, and Pontryagin classes in number theory.

In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the η-Ricci–Yamabe soliton (η-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an η-RY soliton, an η-Ricci soliton, and an η-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an η-RY soliton, an η-Ricci soliton, an η-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field ω of the soliton is of gradient type =:grad(γ) and provide some examples of an η-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers.

In this study, the authors focus on quasi-hemi-slant submanifolds (qhs-submanifolds) of (α,β)-type almost contact manifolds, also known as trans-Sasakian manifolds. Essentially, we give sufficient and necessary conditions for the integrability of distributions using the concept of quasi-hemi-slant submanifolds of trans-Sasakian manifolds. We also consider the geometry of foliations dictated by the distribution and the requirements for submanifolds of trans-Sasakian manifolds with quasi-hemi-slant factors to be totally geodesic. Lastly, we give an illustration of a submanifold with a quasi-hemi-slant factor and discuss its application to number theory.

We propose fundamental inequalities for contact pseudo-slant submanifolds of (ϵ)-para Sasakian space form employing generalized normalized δ-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants.

The aim of the present research is to study the δ-Lorentzian trans Sasakian manifolds with a semi-symmetric metric connection. We have found the expressions for curvature tensors, Ricci curvature tensors and scalar curvature of the δ-Lorentzian trans Sasakian manifolds with a semi-symmetric metric and metric connection. Also, we have discussed some results on quasi-projectively flat and ϕ-projectively flat manifolds endowed with a semi-symmetric-metric connection. It shown that the manifold satisfying ¯ R. ¯ S = 0, ¯ P, ¯ S = 0. Lastly, we have obtained the conditions for the δ-Lorentzian Trans Sasakian manifolds with a semi-symmetric metric connection to be conformally flat and ξ-conformally flat.