Search Results (22)

The subject of this study is almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds. The considerations are restricted to a special class of these manifolds, namely those of the Sasaki-like type, because of their geometric construction and the explicit expression of their classification tensor by the pair of B-metrics. Here, each of the two B-metrics is considered as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form. The soliton potential is chosen to be a conformal Killing vector field (in particular, concircular or concurrent) and then a generalization of the notion of conformality using contact conformal transformations of B-metrics. The resulting manifolds, equipped with the introduced almost solitons, are geometrically characterized. In the five-dimensional case, an explicit example on a Lie group depending on two real parameters is constructed, and the properties obtained in the theoretical part are confirmed. Full article

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

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated $\eta$-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article

Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f-structure of this kind, called the weak $\beta$-Kenmotsu f-structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak $\beta$-Kenmotsu f-manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $\beta =const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature. Full article

The present research paper investigates submanifolds of Kenmotsu manifolds, focusing on those equipped with concurrent vector fields. It examines the structural and geometric properties of such submanifolds, analyzing the decomposed equations in both vertical and horizontal components. Furthermore, the study generalizes certain results in the context of η-Ricci solitons and η-Yamabe solitons. Full article

The interest of geometers in f-structures is motivated by the study of the dynamics of contact foliations, as well as their applications in physics. A weak f-structure on a smooth manifold, introduced by the author and R. Wolak, generalizes K. Yano’s f-structure. This generalization allows us to revisit classical theory and discover applications of Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. This article reviews the results regarding weak metric f-manifolds and their distinguished classes. 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

This paper focuses on some geometrical and physical properties of a conformal η-Ricci soliton (Cη-RS) on a four-dimension Lorentzian Para-Sasakian (LP-S) manifold. The first section presents an introduction to Cη-RS on LP-S manifolds, followed by a discussion of preliminary ideas about the LP-Sasakian manifold. In the subsequent sections, we establish several results pertaining to four-dimension LP-S manifolds that exhibit Cη-RS. Additionally, we consider certain conditions associated with Cη-RS on four-dimension LP-S manifolds. Besides these geometrical points of view, we consider this soliton in a perfect fluid spacetime and obtain some interesting physical properties. Finally, we present a case study of a Cη-RS on a four-dimension LP-S manifold. Full article

In this research article, we concentrate on the exploration of submanifolds in an ${\left(LCS\right)}_m$-manifold $\tilde{B}$. We examine these submanifolds in the context of two distinct vector fields, namely, the characteristic vector field and the concurrent vector field. Initially, we consider some classifications of $\eta$-Ricci–Bourguignon (in short, $\eta$-RB) solitons on both invariant and anti-invariant submanifolds of $\tilde{B}$ employing the characteristic vector field. We establish several significant findings through this process. Furthermore, we investigate additional results by using $\eta$-RB solitons on invariant submanifolds of $\tilde{B}$ with concurrent vector fields, and discuss a supporting example. Full article

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 $\eta$-Ricci solitons ($\eta$-RS) for an interesting manifold called the $(\epsilon )$-Kenmotsu manifold ($(\epsilon )$-$KM$), endowed with a semi-symmetric metric connection (briefly, a SSM-connection). We discuss Ricci and $\eta$-Ricci solitons with a SSM-connection satisfying certain curvature restrictions. In addition, we consider the characteristics of the gradient $\eta$-Ricci solitons (a special case of $\eta$-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 $\eta$-Ricci solitons for $(\epsilon )$-Kenmotsu manifold, with a semi-symmetric metric connection. Finally, we explore a number theoretic approach in the form of Pontrygin numbers to the $(\epsilon )$-Kenmotsu manifold equipped with a semi-symmetric metric connection. Full article

In this manifestation, we explain the geometrisation of $\eta$-Ricci–Yamabe soliton and gradient $\eta$-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 $\eta$-Ricci–Yamabe soliton, which is called the solitonic fiber. Also, under the same setting, we inspect the $\eta$-Ricci–Yamabe soliton in Riemannian submersions with a $\phi (\mathcal{Q})$-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. Full article

In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the $\eta$-Ricci–Yamabe soliton ($\eta$-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an $\eta$-RY soliton, an $\eta$-Ricci soliton, and an $\eta$-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an $\eta$-RY soliton, an $\eta$-Ricci soliton, an $\eta$-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 $\omega$ of the soliton is of gradient type =:grad$(\gamma )$ and provide some examples of an $\eta$-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers. Full article

We obtain on a Kähler B-manifold (i.e., a Kähler manifold with a Norden metric) some corresponding results from the Kählerian and para-Kählerian context concerning the Bochner curvature. We prove that such a manifold is of constant totally real sectional curvatures if and only if it is a holomorphic Einstein, Bochner flat manifold. Moreover, we provide the necessary and sufficient conditions for a gradient Ricci soliton or a holomorphic $\eta$-Einstein Kähler manifold with a Norden metric to be Bochner flat. Finally, we show that a Kähler B-manifold is of quasi-constant totally real sectional curvatures if and only if it is a holomorphic $\eta$-Einstein, Bochner flat manifold. Full article

The present paper deals with the investigations of a Kenmotsu manifold satisfying certain curvature conditions endowed with $★$-η-Ricci solitons. First we find some necessary conditions for such a manifold to be φ-Einstein. Then, we study the notion of $★$-η-Ricci soliton on this manifold and prove some significant results related to this notion. Finally, we construct a nontrivial example of three-dimensional Kenmotsu manifolds to verify some of our results. Full article

This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal $\eta$-Ricci soliton and gradient conformal $\eta$-Ricci soliton with a potential vector field $\zeta$. Additionally, we estimate the various conditions for which the target manifold of Riemannian submersion under the canonical variation is a conformal $\eta$-Ricci soliton with a Killing vector field and a $\phi \left(\mathcal{R}ic\right)$-vector field. Moreover, we deduce the generalized Liouville equation for Riemannian submersion under the canonical variation satisfying by a last multiplier $\mathsf{\Psi }$ of the vertical potential vector field $\zeta$ and show that the base manifold of Riemanian submersion under canonical variation is an $\eta$ Einstein for gradient conformal $\eta$-Ricci soliton with a scalar concircular field $\gamma$ on base manifold. Finally, we illustrate an example of Riemannian submersions between Riemannian manifolds, which verify our results. Full article