Search Results (16)

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

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique $\eta$-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang. 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

We study the quarter-symmetric metric A-connection on a cosymplectic manifold. Observing linearly independent curvature tensors with respect to the quarter-symmetric metric A-connection, we construct the Weyl projective curvature tensor on a cosymplectic manifold. In this way, we obtain new conditions for the manifold to be projectively flat. At the end of the paper, we define $\eta$-Einstein cosymplectic manifolds of the $\theta$-th kind and prove that they coincide with the $\eta$-Einstein cosymplectic manifold. 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

Almost-Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e., an almost-contact B-metric manifold, which is obtained from a cosymplectic manifold of the considered type by means of a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The potential of the studied soliton is assumed to be in the vertical distribution, i.e., it is collinear to the Reeb vector field. In this way, manifolds from the four main classes of the studied manifolds are obtained. The curvature properties of the resulting manifolds are derived. An explicit example of dimension five is constructed. The Bochner curvature tensor is used (for a dimension of at least seven) as a conformal invariant to obtain these properties and to construct an explicit example in relation to the obtained 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

In this paper, we study the properties of $ϵ$-Kenmotsu manifolds if its metrics are $*\eta$-Ricci-Yamabe solitons. It is proven that an $ϵ$-Kenmotsu manifold endowed with a $*\eta$-Ricci-Yamabe soliton is $\eta$-Einstein. The necessary conditions for an $ϵ$-Kenmotsu manifold, whose metric is a $*\eta$-Ricci-Yamabe soliton, to be an Einstein manifold are derived. Finally, we model an indefinite Kenmotsu manifold example of dimension 5 to examine the existence $*\eta$-Ricci-Yamabe solitons. Full article

Para-Ricci-like solitons with arbitrary potential on para-Sasaki-like Riemannian $\Pi$-manifolds are introduced and studied. For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the corresponding scalar curvatures of both considered metrics are equal and constant. An explicit example of the Lie group as the manifold under study is presented. Full article

A submersion from an almost contact Riemannian manifold to an almost Hermitian manifold, acting on the horizontal distribution by preserving both the metric and the structure, is, roughly speaking a contact-complex Riemannian submersion. This paper deals mainly with a contact-complex Riemannian submersion from an $\eta$-Ricci soliton; it studies when the base manifold is Einstein on one side and when the fibres are $\eta$-Einstein submanifolds on the other side. Some results concerning the potential are also obtained here. 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