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Purpose: In 1979, P. Wintgen obtained a basic relationship between the extrinsic normal curvature the intrinsic Gauss curvature, and squared mean curvature of any surface in a Euclidean 4-space with the equality holding if and only if the curvature ellipse is a circle. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture of Wintgen inequality, named as the DDVV-conjecture, for general Riemannian submanifolds in real space forms. Later on, this conjecture was proven to be true by Z. Lu and by Ge and Z. Tang independently. Since then, the study of Wintgen's inequalities and Wintgen ideal submanifolds has attracted many researchers, and a lot of interesting results have been found during the last 15 years. The main purpose of this paper is to extend this conjecture of Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection. Design/methodology/approach: The authors used standard technique for obtaining generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection. Findings: The authors establish the generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection, and also find conditions under which the equality holds. Some particular cases are also stated. Originality/value: The research may be a challenge for new developments focused on new relationships in terms of various invariants, for different types of submanifolds in that ambient space with several connections. [ABSTRACT FROM AUTHOR]

This work aims to provide generalized Wintgen inequalities for slant submanifolds embedded in quaternionic space forms, taking into consideration both semi-symmetric metric and semi-symmetric non-metric connections. Moreover, we discuss the same inequality for totally real, anti-invariant, and invariant submanifolds on quaternionic space forms, endowed with both semi-symmetric metric and semi-symmetric non-metric connections. We also characterized the equality case through specific forms of shape operators of Wintgen inequalities for these classes of submanifolds in quaternionic space forms, admitting a semi-symmetric metric connection and a semi-symmetric non-metric connection. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the present research note, we discuss the energy–momentum squared gravity model F (R , T 2) coupled with perfect fluid. We obtain the equation of state for the perfect fluid in the F (R , T 2) -gravity model. Furthermore, we deal with the energy–momentum squared gravity model F (R , T 2) 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 , T 2) -gravity in terms of hyperbolic Ricci solitons. Lastly, we deduce the Schrödinger equation for the potential U n with hyperbolic Ricci solitons in the F (R , T 2) -gravity model coupled with perfect fluid and a phantom barrier. [ABSTRACT FROM AUTHOR]

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 '(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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this research paper, we discussed some geometric axioms of a relativistic string cloud spacetime attached with strange quark matter. We determined the conformal η-Ricci soliton on a relativistic string cloud spacetime attached with strange quark matter with a φ(Ric)-vector field. In addition, we illustrated some physical significance of conformal pressure P in terms of conformal η-Ricci soliton with the same vector field. Besides this, we deduced a generalized Liouville equation from the conformal η-Ricci soliton. Furthermore, we examine the harmonic relevance of conformal η-Ricci soliton on string cloud spacetime attached with strange quark matter with a potential function ψ. Finally, we turned up a necessary and sufficient condition for the 1-form η, which is the g-dual of the vector field γ on a string cloud spacetime attached with strange quark matter to be a solution for the Schrödinger-Ricci equation. [ABSTRACT FROM AUTHOR]

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 ( S V K -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. [ABSTRACT FROM AUTHOR]

The purpose of the article is to analyze the behavior of spacetime using a string cloud energy-momentum tensor T having string cloud fluid density ρ and string tension λ, 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. [ABSTRACT FROM AUTHOR]

In this study, the authors focus on quasi-hemi-slant submanifolds ( q h s -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. [ABSTRACT FROM AUTHOR]

The target of the current research article is to investigate the solitonic attributes of relativistic magneto-fluid spacetime (MFST) if its metrics are Ricci–Yamabe soliton (RY-soliton) and gradient Ricci–Yamabe soliton (GRY-soliton). We exhibit that a magneto-fluid spacetime filled with a magneto-fluid density ρ , magnetic field strength H, and magnetic permeability μ obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold (G Q E) . In such a spacetime, we obtain an EoS with a constant scalar curvature R in terms of the magnetic field strength H and magnetic permeability μ. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field ξ and a φ (R i c) vector field. We establish the existence of a black hole in the relativistic magneto-fluid spacetime by demonstrating that it admits a shrinking Ricci–Yamabe soliton and satisfies the time-like energy convergence criteria. In addition, we examine the magneto-fluid spacetime with a gradient Ricci–Yamabe soliton and deduce some conditions for an equation of state (EoS) ω = − 1 5 with a Killing vector field. Furthermore, we demonstrate that the EoS ω = − 1 5 of the magneto-fluid spacetime under some constraints represents a star model and a static, spherically symmetric perfect fluid spacetime. Finally, we prove that a gradient Ricci–Yamabe soliton with the conditions μ = 0 or H = 2 ; μ ≠ 0 , H > 2 and obeying the equation of state ω = − 1 5 is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively. [ABSTRACT FROM AUTHOR]

The present research article is concerned about a couple of optimal inequalities for submanifolds of δ-Lorentzian trans-Sasakian manifolds endowed with semi-symmetric metric connection (briey says SSM). Some examples of δ-Lorentzian trans-Sasakiam manifolds are also discussed here. This paper ends with some open problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This research article attempts to examine the attribute of perfect fluid spacetime in f(R, T) gravity with a Killing velocity vector field ρ in terms of Ricci soliton, gradient Ricci soliton, Yamabe soliton, and gradient Yamabe soliton. Besides this, we evaluate a specific situation when the potential vector field ρ is of the form of gradient, we extract a modified Poisson equation, and modified Liouville equation from the Ricci soliton equation in f(R, T)-gravity stuffing with perfect fluids. In addition, we explore some harmonic significance of Ricci soliton on perfect fluid spacetime in f(R, T) gravity with a harmonic potential function Ψ. [ABSTRACT FROM AUTHOR]

The objective of this study is to investigate η-Einstein solitons on (ε)-Kenmotsu manifolds when the Weyl-conformal curvature tensor satisfies some geometric properties such as being flat, semi-symmetric and Einstein semi-symmetric. Here, we discuss the properties of η-Einstein solitons on φ-symmetric (ε)-Kenmotsu manifolds. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the properties of three dimensional Lorentzian concircular structure manifolds ((LCS)3-manifolds) endowed with almost h-conformal Ricci solitons. Also, we discuss the h-conformal gradient shrinking Ricci solitons on (LCS)3-manifolds. Finally, the examples of almost h-conformal Ricci soliton on an (LCS)3-manifold are provided in the region where (LCS)3-manifold is expanding. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the δ-Lorentzian trans-Sasakian manifold endowed with semi-symmetric metric connections admitting η-Ricci Solitons and Ricci Solitons. We find expressions for the curvature tensor, the Ricci curvature tensor and the scalar curvature tensor of δ-Lorentzian trans-Sasakian manifolds with a semi-symmetric-metric connection. Also, we discuses some results on quasi-projectively flat and ϕ-projectively flat manifolds endowed with a semi-symmetric-metric connection. It is shown that the manifold satisfying R̅.S̅ = 0, P̅.S̅ = 0 is an η-Einstein manifold. More-over, we obtain the conditions for the δ-Lorentzian trans-Sasakian manifolds with a semi-symmetric-metric connection to be conformally flat and ξ-conformally flat. [ABSTRACT FROM AUTHOR]

f (R , T) -gravity is a generalization of Einstein's field equations (E F E s) and f (R) -gravity. In this research article, we demonstrate the virtues of the f (R , T) -gravity model with Einstein solitons (E S) and gradient Einstein solitons (G E S) . 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. [ABSTRACT FROM AUTHOR]