Search Results (103)

This paper examines the classification of Lorentzian hypersurfaces in Minkowski space, specifically focusing on those that are characterized as good or bad and that admit Ricci solitons with a concircular vector field. First, we demonstrate that Ricci solitons of this nature do not exist in Lorentzian spaces that have non-zero sectional curvature. Additionally, we show that the shape operator can be expressed in specific canonical forms. Full article

In this article, we are interested in finding sufficient conditions for a noncompact Ricci soliton (M, h, v, λ) to be trivial. The major role is played by a skew-symmetric operator φ associated to the potential field v of the connected Ricci soliton (M, h, v, λ). In the firstresult, it is shown that if the potential field v of the Ricci soliton (M, h, v, λ) is an eigenvector of the Ricci operator S with eigenvalue τ/n, where τ is the scalar curvature and n = dim M, with additional conditions v(τ) ≤ 0 and squared norm of the covariant derivative ∇v having a suitable upper bound is necessarily trivial. In another result, it is shown that a connected Ricci soliton (M, h, v, λ) with potential field v an eigenvector of the Ricci operator S with eigenvalue λ with an additional condition v(τ) ≤ 0 is trivial. Similarly, it is shown that for a complete and connected Ricci soliton (M, h, v, λ) with potential field v annihilating the Ricci operator S and incompressible vector field φv and v(τ) having a suitable upper bound is either trivial, or else is isometric to the Euclidean space. Also, we show that if the Ricci operator S of a connected Ricci soliton (M, h, v, λ) is invariant under the potential field v with additional two suitable conditions, then (M, h, v, λ) is trivial. Finally, it has been observed that the function ψ related to the squared length of the potential field v of a connected Ricci soliton (M, h, v, λ) has a role, namely it is shown that if ψ is a superharmonic function and the Ricci curvature Ric(v, v) has a suitable upper bound then (M, h, v, λ) is necessarily trivial. Full article

In this article, some remarkable results on general relativistic spacetimes with non-constant scalar curvature $\tau$, admitting almost Ricci-Bourguignon solitons and gradient almost Ricci-Bourguignon solitons, have been established. Finally, a non-trivial example of a general relativistic spacetime is constructed by using partial differential equations to validate some of our findings. Full article

This manuscript investigates conformal $\eta$-Ricci–Yamabe solitons of type $(\kappa ,l)$ on Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field. Necessary conditions are obtained under which the soliton exhibits expanding, steady, or shrinking behavior. The analysis is further extended to several physically relevant fluid models, including dark fluid, dust fluid, stiff matter, and radiational fluid, and the corresponding geometric constraints are derived. In addition, structural results are established for Kählerian Norden space-times with a vanishing space–matter tensor and with a divergence-free matter tensor, highlighting their influence on the curvature geometry. The study also addresses several intrinsic curvature conditions of the space-time, such as conformal flatness, Ricci semi-symmetry, Ricci recurrence, and pseudo-Ricci symmetry, leading to a collection of geometric and physical characterizations. The results obtained provide a unified geometric framework linking Ricci–Yamabe soliton structures, fluid dynamics, and curvature properties within the setting of Kählerian Norden geometry. Full article

We consider Riemann soliton vector fields and Ricci bi-conformal vector fields on the oscillator group. We prove that the oscillator group admits Riemann solitons. Subsequently, we provide a complete classification of all Ricci bi-conformal vector fields admitted by the oscillator group and identify those that belong to specific categories, namely gradient-type vector fields, Killing vector fields, and Ricci collineation using the partial differential equations. Full article

This article investigates the geometric and structural properties of almost pure metric pseudo-F-manifolds, with a focus on Walker 4-manifolds. We analyze integrability conditions and characterize pure metric pseudo-F-Kählerian Walker manifolds, identifying criteria under which the Riemann curvature tensor vanishes. The study also examines the existence of Killing vector fields and Ricci soliton structures. In particular, we show that integrability conditions for almost pure metric pseudo-F-structures are governed by partial differential equations and highlight the role of constant functions in defining pure metric pseudo-F-Kählerian properties. Additionally, we investigate special connections with torsion that preserve certain tensor structures, exploring their relationship to Codazzi pairs and the conditions necessary for torsion-freeness. Full article

We investigate Ricci–Bourguignon solitons on mixed doubly sequential warped product manifolds. Necessary and sufficient conditions for the existence of such solitons are established, and their implications for Einstein manifolds are analyzed. This work extends previous results on warped product manifolds to the more complex case of mixed doubly sequential warped products. We derive explicit formulas for Ricci tensor components, establish conditions for Einstein metrics, and study gradient solitons and conformal vector fields. Our classification theorems reveal fundamental separability constraints on the warping functions. Applications to cosmological models and black hole solutions demonstrate the physical relevance of these results. Full article

The present paper is aimed at studying the convergence of the Yamabe flow in the case of noncompact solitons. The more specified example of locally conformally flat noncompact solitons is addressed with the aim to newly analyse the qualities of the Ricci scalar. The particular case of noncompact pseudo-Riemannian solitons is studied; moreover, in the instances of Schwarzschild and Generalized-Schwarzschild geometries, rescalings of spherically symmetric weights are performed. For this purpose, new results are achieved as far as the considered structures are concerned. The Myers Theorem is upgraded as the new Myers paradigm of spacetime-dimensional manifolds, where the Einstein Field Equations can now be taken into account. In particular, the Myers Theorems are studied here as far as their new implementation in General Relativity Theory is concerned. As a first important result, the Myers mean curvature is found to coincide with the Ricci scalar in General Relativity Theory, where the 4-position of the observer, from which the 4-velocity 4-vector is calculated from, is taken as that of the observer solidal with the reference frame of the photon. The following results are also of relevance. In more detail, the umbilicity conditions are applied. At a further step, the role of the umbilicity conditions in GR after the Myers Theorems are studied for weighted manifolds and specific new implications of weighted manifolds are developed. The description of the weighted Schwarzschild manifolds and that of the weighted Generalized-Schwarzschild manifolds are newly studied as follows: as a new finding, the Birkhoff Theorem is newly reconciled with the rotational ansatz of the metrised solitons, and the comparison with the previous results about the Brendle non-metrised solitons is accomplished with the outcome stressing the new roles of the new rescalings of the metric tensor with respect to the previous known results of the scaling of the metric tensor of the non-metrised solitons. In the present framework, these procedures allow one to prove the reconciliation of the EFEs with the Yamabe flow. The flow on the tipping lightcones is newly written. The umbilicity condition is studied in General Relativity after the upgrade of the Myers Theorems as far as the sectional curvatures are concerned; as a result, the Calabi–Bernstein description is implemented in General Relativity, as well as the Chen–Yau requirements, and the cases of weighted manifolds are taken into account. More specifically, the equal-time 2-dimensional space surfaces are studied analytically, onto which the weighted General-Relativistic solitons which satisfy the Einstein field equations after the Yamabe flow are projected due to the rotational ansatz. As an accessory introductory result, the class of Wu non-metrised solitons are proven to be discarded in several aspects of the Wu description as the conditions provided after the work of Wu are not compatible with metrisation. Full article

This study is dedicated to a separable $F(\mathfrak{R},\mathfrak{T})$-gravity related to the anisotropic matter to extract the equation of state for $F(\mathfrak{R},\mathfrak{T})$-gravity. In this research, we offer insight into calculating the density and pressure in the phantom barrier, stiff fluid, and matter-dominated eras, respectively. As demonstrated, a spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity full of anisotropic matter is a generalized quasi-Einstein spacetime. In addition, we gain the equation of state of Codazzi type, Ricci semi-symmetric and Ricci-pseudo symmetric anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity. We prove an anisotropic spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity endowed with Codazzi-type Ricci tensor is a Yang Pure spacetime and Robertson–Walker spacetime. Furthermore, we try to give out the energy constraints of Penrose’s singularity theorem for black holes in an anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity. Lastly, we study hyperbolic Ricci solitons on anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity endowed with a torse-forming vector field, and for steady hyperbolic Ricci soliton, we deduced the equation of state of anisotropic fluid spacetime in $F(\mathfrak{R},\mathfrak{T})$-gravity. Full article

This manuscript presents a comprehensive taxonomy of Riemann solitons within the framework of a spacetime manifold endowed with a metric exhibiting both spatial homogeneity and rotational characteristics. Furthermore, we undertake an analysis to determine the geometric nature of these solitons by establishing their correspondence to Killing vector fields, Ricci collineation vector fields, and gradient vector fields. Full article

In this paper, we examine Ricci–Bourguignon solitons on locally decomposable golden Riemannian manifolds of constant golden sectional curvature. First, we establish an explicit expression for the soliton constant in terms of the golden structure and the Bourguignon parameter. Second, we explore the geometry of these solitons when the potential vector fields are Killing, conformal Killing, homothetic, or concurrent. Finally, we initiate the study of golden Ricci–Bourguignon solitons, determine their soliton constants, and examine their properties under the specific potential vector fields. Full article

In this article, we study the Ricci soliton on quaternion bi-slant submanifolds of quaternion Kaehler manifolds. We obtain a lower-bound-type inequality in terms of expanding gradient Ricci solitons with a gradient-type vector field for the quaternion bi-slant submanifold of quaternion Kaehler manifolds. Additionally, we derive a series of lower-bound-type inequalities for semi-slant submanifolds, $CR$-submanifolds, hemi-slant submanifolds, slant submanifolds, invariant anti-invaraint submanifolds and totally real submanifolds in the same quaternion Kaehler manifolds. Finally, we discuss a double inequality for expanding gradient Ricci solitons on submanifolds in quaternion Kaehler manifolds and extend the same double inequality in terms of gradient Ricci solitons with a scalar concircular field on semi-slant, quaternion $CR$-submanifolds of quaternion Kaehler manifolds. Full article

In this paper we investigate Killing and 2-Killing 1-forms on a Poisson doubly warped product manifold ($\mathcal{PDWPM}$). We establish a connection between the property of 1-form on a $\mathcal{PDWPM}$ and its components on the factor manifolds to be Killing or 2-Killing. We also demonstrate that a Killing 1-form on a $\mathcal{PDWPM}$ generates a contravariant Ricci soliton factor manifold if and only if it is a contravariant Einstein manifold. Additionally, we provide necessary and sufficient conditions for a $\mathcal{PDWPM}$ to be a Poisson singly warped product manifold ($\mathcal{PSWPM}$). Finally, we present examples of Killing and 2-Killing 1-forms on $\mathcal{PDWPM}s$ with one-dimensional, and two-dimensional base spaces. Full article

Each of the studied manifolds has a pair of B-metrics, interrelated by an almost contact structure. The case where each of these metrics gives rise to an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, is studied. In addition, the geometry-rich case where the soliton potential is torse-forming and is pointwise collinear on the Reeb vector field with respect to each of the two metrics is considered. Ricci tensors and scalar curvatures are expressed as functions of the parameters of the pair of almost solitons. Particular attention is paid to the special case when the manifold belongs to the only possible basic class of the corresponding classification. A necessary and sufficient condition has been found for these almost solitons to be $\eta$-Einstein for both metrics. Full article

In this research paper, we introduce the notions of hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons. We study the hyperbolic ∗-Ricci solitons on a three-dimensional $\epsilon$-trans-Sasakian manifold. Specifically, we determine the hyperbolic ∗-Ricci solitons on a three-dimensional ($\epsilon$)-trans-Sasakian manifold with a conformal vector field and a proper $\varphi \left({\mathfrak{Q}}^{*}\right)$-type vector field. Using hyperbolic ∗-Ricci solitons with a conformal vector field, we discuss some geometric symmetries on a three-dimensional ($\epsilon$)-trans-Sasakian. In addition, we exhibit the nature of gradient hyperbolic ∗-Ricci solitons on a three-dimensional ($\epsilon$)-trans-Sasakian manifold endowed with a scalar concircular field. Moreover, we demonstrate an example on a three-dimensional ($\epsilon$)-trans-Sasakian manifold that admits the hyperbolic ∗-Ricci solitons and find the rate of change of the hyperbolic ∗-Ricci solitons within the same example. Lastly, we also introduce the concept of modified second hyperbolic ∗-Ricci solitons. Full article