Search Results (31)

In this paper, we investigate contact skew CR-warped product submanifolds of locally conformal almost cosymplectic manifolds, a framework that simultaneously generalizes warped product pseudo-slant, semi-slant, and contact CR-submanifolds. We first establish a necessary and sufficient characterization theorem showing that a proper contact skew CR-submanifold with integrable slant distribution admits a local warped product structure if and only if certain shape operator conditions involving the slant angle and the warping function are satisfied. Subsequently, we derive sharp geometric inequalities for the squared norm of the second fundamental form in terms of the warping function, the slant angle, and the conformal factor of the ambient manifold. The equality cases are completely characterized and lead to strong rigidity results, namely that the base manifold is totally geodesic while the slant fiber is totally umbilical in the ambient space. Several applications are presented, showing that our results recover and extend a number of known inequalities and classification theorems for warped product submanifolds in cosymplectic, Kenmotsu, and Sasakian geometries as special cases. Full article

The main subject of this paper is the theme of differential operators defined for symmetric tensors on a Riemannian manifold and introduced in several new contexts. Some examples for 2-tensors are given and then the grad div operator for symmetric vector forms is defined. A few original operators in ${\mathbb{R}}^n$ related with grad div operator are discussed. Finally, important notions such as the Kenmotsu manifold, with some interesting examples, are also presented. Full article

This research paper aims to study the Q-curvature tensor on Kenmotsu manifolds endowed with the Schouten–van Kampen connection. Using the Q-curvature tensor, whose trace is the well-known Z-tensor, we characterized Kenmotsu manifolds by introducing the notion of $\zeta$-$\tilde{Q}$ flat and $\varphi$-$\tilde{Q}$ flat manifolds and novel tensor conditions, such as $\tilde{Q}(\xi ,X)\tilde{Q}=0,$$\tilde{Q}(\xi ,X)\tilde{R}=0,$$\tilde{Q}(\xi ,X)\tilde{C}=0,$$\tilde{Q}(\xi ,X)\tilde{S}=0$, $\tilde{Q}(\xi ,X)\tilde{H}=0$, and $\tilde{Q}(\xi ,X)\tilde{P}=0$, with the Schouten–van Kampen connection. To validate some of our results, we constructed a non-trivial example of Kenmotsu manifolds endowed with the Schouten–van Kampen connection. 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

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

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in different areas of differential geometry, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as Golden space forms, Sasakian space forms; diffeological and affine connection spaces; Weingarten and Delaunay surfaces; Chen-type inequalities for submanifolds; statistical submersions; manifolds endowed with different geometric structures (Sasakian, weak nearly Sasakian, weak nearly cosymplectic, LP-Kenmotsu, paraquaternionic); solitons (almost Ricci solitons, almost Ricci–Bourguignon solitons, gradient r-almost Newton–Ricci–Yamabe solitons, statistical solitons, solitons with semi-symmetric connections); vector fields (projective, conformal, Killing, 2-Killing) [...] Full article

The focus of this research is on investigating a new category of generalized symmetric metric connections within nearly Kenmotsu manifolds. The study delves into recognizing the generalized symmetric connections of type $(\alpha , \beta )$, which represent broader versions of the semi-symmetric metric connection $(\alpha =1, \beta =0)$ and the quarter-symmetric metric connection $(\alpha =0, \beta =1)$. 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

In the current article, we examine Lorentzian para-Kenmotsu (shortly, LP-Kenmotsu) manifolds with regard to the generalized symmetric metric connection ${\nabla }^{\mathcal{G}}$ of type $(\alpha ,\beta )$. First, we obtain the expressions for curvature tensor, Ricci tensor and scalar curvature of an LP-Kenmotsu manifold with regard to the connection ${\nabla }^{\mathcal{G}}$. Next, we analyze LP-Kenmotsu manifolds equipped with the connection ${\nabla }^{\mathcal{G}}$ that are locally symmetric, Ricci semi-symmetric, and $\phi$-Ricci symmetric and also demonstrated that in all these situations the manifold is an Einstein one with regard to the connection ${\nabla }^{\mathcal{G}}$. Moreover, we obtain some conclusions about projectively flat, projectively semi-symmetric and $\phi$-projectively flat LP-Kenmotsu manifolds concerning the connection ${\nabla }^{\mathcal{G}}$ along with several consequences through corollaries. Ultimately, we provide a 5-dimensional LP-Kenmotsu manifold example to validate the derived expressions. Full article

In this article, we delve into the study of statistical solitons on submanifolds of Kenmotsu statistical manifolds, introducing the presence of concircular vector fields. This investigation is further extended to study the behavior of almost quasi-Yamabe solitons on submanifolds with both concircular and concurrent vector fields. Concluding our research, we offer a compelling example featuring a 5-dimensional Kenmotsu statistical manifold that accommodates both a statistical soliton and an almost quasi-Yamabe soliton. This example serves to reinforce and validate the principles discussed throughout our study. Full article

The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We also study and discover that the lift of the Kenmotsu manifold that admit NSNMC is regular in the tangent bundle. Additionally, we find that the data provided by the lift of Ricci soliton on the lift of Ricci semi-symmetric Kenmotsu manifold that admits NSNMC in the tangent bundle are expanding. Full article

The main objective of this paper is to study semi-symmetric almost $\alpha$-cosymplectic three-manifolds. We present basic formulas for almost $\alpha$-cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost $\alpha$-cosymplectic three-manifolds. We obtain the main results under an additional condition. The paper concludes with two illustrative examples. 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