Search Results (14)

This paper undertakes a detailed study of $\eta$-Ricci–Bourguignon solitons on $ϵ$-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic $\eta$-recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying Ricci semi-symmetric $(R·E=0)$, conharmonically Ricci semi-symmetric $(C(\xi ,{\beta }_X)·E=0)$, $\xi$-projectively flat $(P({\beta }_X,{\beta }_Y)\xi =0)$, projectively Ricci semi-symmetric $(L·P=0)$ and $W_5$-Ricci semi-symmetric $(W(\xi ,{\beta }_Y)·E=0)$, respectively, with the admittance of $\eta$-Ricci–Bourguignon solitons. This work further explores the role of torse-forming vector fields and provides a thorough characterization of $\varphi$-Ricci symmetric indefinite Kenmotsu manifolds admitting $\eta$-Ricci–Bourguignon solitons. Through in-depth analysis, we establish significant geometric constraints that govern the behavior of these manifolds. Finally, we construct explicit examples of indefinite Kenmotsu manifolds that satisfy the $\eta$-Ricci–Bourguignon solitons equation, thereby confirming their existence and highlighting their unique geometric properties. Moreover, these solitonic structures extend soliton theory to indefinite and physically meaningful settings, enhance the classification and structure of complex geometric manifolds by revealing how contact structures behave under advanced geometric flows and link the pure mathematical geometry to applied fields like general relativity. Furthermore, $\eta$-Ricci–Bourguignon solitons provide a unified framework that deepens our understanding of geometric evolution and structure-preserving transformations. Full article

In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and Lorentzian manifolds. First, we analyze the properties of these vector fields on Riemannian manifolds. Next, we focus on Ricci solitons on Riemannian hypersurfaces induced by torse-forming vector fields of Riemannian or Lorentzian manifolds. Specifically, we show that such a hypersurface in the manifold with constant sectional curvature is either totally geodesic or an extrinsic sphere. Full article

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are equipped with Ricci–Bourguignon-like almost solitons. These almost solitons are a generalization of the known Ricci–Bourguignon almost solitons, in which, in addition to the main metric, the associated metric of the manifold is also involved. In the present paper, the soliton potential is specialized to be pointwise collinear with the Reeb vector field of the manifold structure, as well as torse-forming with respect to the two Levi-Civita connections of the pair of B-metrics. The forms of the Ricci tensor and the scalar curvatures generated by the pair of B-metrics on the studied manifolds with the additional structures have been found. In the three-dimensional case, an explicit example is constructed and some of the properties obtained in the theoretical part are illustrated. Full article

We highlight some properties of a class of distinguished vector fields associated to a $(1,1)$-tensor field and to an affine connection on a Riemannian manifold, with a special view towards the Ricci vector fields, and we characterize them with respect to statistical, almost Kähler, and locally product structures. In particular, we provide conditions for these vector fields to be closed, Killing, parallel, or semi-torse forming. In the gradient case, we give a characterization of the Euclidean sphere. Among these vector fields, the Ricci and torse-forming-like vector fields are particular cases. Full article

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are, in principle, equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e., its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e., it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi–Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed, and the properties obtained in the theoretical part are confirmed. Full article

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 $\rho$, magnetic field strength H, and magnetic permeability $\mu$ obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold $\left(GQE\right)$. In such a spacetime, we obtain an EoS with a constant scalar curvature $\mathcal{R}$ in terms of the magnetic field strength H and magnetic permeability $\mu$. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field $\xi$ and a $\phi \left(\mathcal{R}ic\right)$ 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) $\omega =-\frac{1}{5}$ with a Killing vector field. Furthermore, we demonstrate that the EoS $\omega =-\frac{1}{5}$ 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 $\mu =0$ or $H=2$; $\mu \ne 0$, $H>2$ and obeying the equation of state $\omega =-\frac{1}{5}$ is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively. Full article

A Yamabe soliton is considered on an almost-contact complex Riemannian manifold (also known as an almost-contact B-metric manifold), which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. A case in which the potential is a torse-forming vector field of constant length on the vertical distribution determined by the Reeb vector field is studied. In this way, manifolds from one of the main classes of the studied manifolds are obtained. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the given B-metric. An explicit five-dimensional example of a Lie group is given, which is characterized in relation to the obtained results. Full article

The present research paper consists of the study of an ${\eta }_1$-Einstein soliton in general relativistic space-time with a torse-forming potential vector field. Besides this, we try to evaluate the characterization of the metrics when the space-time with a semi-symmetric energy-momentum tensor admits an ${\eta }_1$-Einstein soliton, whose potential vector field is torse-forming. In adition, certain curvature conditions on the space-time that admit an ${\eta }_1$-Einstein soliton are explored and build up the importance of the Laplace equation on the space-time in terms of ${\eta }_1$-Einstein soliton. Lastly, we have given some physical accomplishment with the connection of dust fluid, dark fluid and radiation era in general relativistic space-time admitting an ${\eta }_1$-Einstein soliton. Full article

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold $\left(M,g,f,\mu \right)$ to be a generalized Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a $\left(\lambda ,n+m\right)$-Einstein manifold $\left(M,g,w\right)$ having harmonic Weyl tensor, $\left({\nabla }^{j}w\right)\left({\nabla }^{m}w\right){\mathcal{C}}_{jklm}=0$ and ${\nabla }_{l}w{\nabla }^{l}w<0$ reduces to a perfect fluid generalized Robertson–Walker spacetime with Einstein fibers. Finally, $\left(M,g,w\right)$ reduces to a perfect fluid manifold if $\phi =-m\nabla \left(\ln w\right)$ is a $\phi \left(Ric\right)$-vector field on M and to an Einstein manifold if $\psi =\nabla w$ is a $\psi \left(Ric\right)$-vector field on M. Some consequences of these results are considered. Full article

The present paper aims to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field $\xi$ in connection with conformal Ricci–Yamabe metric and conformal $\eta$-Ricci–Yamabe metric. We delineate the conditions for conformal Ricci–Yamabe soliton to be expanding, steady or shrinking. We also discuss conformal Ricci–Yamabe soliton on some special types of perfect fluid spacetime such as dust fluid, dark fluid and radiation era. Furthermore, we design conformal $\eta$-Ricci–Yamabe soliton to find its characteristics in a perfect fluid spacetime and lastly acquired Laplace equation from conformal $\eta$-Ricci–Yamabe soliton equation when the potential vector field $\xi$ of the soliton is of gradient type. Overall, the main novelty of the paper is to study the geometrical phenomena and characteristics of our newly introduced conformal Ricci–Yamabe and conformal $\eta$-Ricci–Yamabe solitons to apply their existence in a perfect fluid spacetime. Full article

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 $\xi$ with potential function $\rho$ on a Lorentzian manifold $(M,g)$, dim$M>5$, 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 $\xi \left(\rho \right)$ 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 $\xi$, there is a function f called the associated function of $\xi$. It is shown that if a connected Lorentzian manifold $(M,g)$, dim$M>4$, admits a time-like special torse-forming vector field $\xi$ with associated function f nowhere zero and satisfies the Fischer–Marsden equation, then $(M,g)$ is a quasi-Einstein manifold. Full article

A torqued vector field $\xi$ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative. Full article

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form $f(z)=b{(z+d)}^{-1}$. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields. Full article

We introduce and study a new type of soliton with a potential Reeb vector field on Riemannian manifolds with an almost paracontact structure corresponding to an almost paracomplex structure. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field were considered. It was proved a necessary and sufficient condition for the manifold to admit a para-Ricci-like soliton, which is the structure that is para-Einstein-like. Explicit examples are provided in support of the proven statements. Full article