Search Results (15)

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 introduce and conduct a comprehensive study of pseudo-Schouten symmetric manifolds ${\left(PSS\right)}_n$. We establish necessary and sufficient conditions for such a manifold to be Einstein and quasi-Einstein, respectively. Next, we examine pseudo-Schouten symmetric spacetimes within the framework of general relativity. Furthermore, we investigate their role in relativistic spacetime models by considering Einstein’s field equations with and without a cosmological constant. We also show that pseudo-Schouten symmetric spacetimes satisfying Einstein’s equations with a quadratic Killing energy–momentum tensor or a Codazzi-type energy–momentum tensor cannot have non-zero constant scalar curvature. Finally, the existence of pseudo-Schouten symmetric spacetime is shown by constructing an explicit non-trivial example. Full article

This paper deals with statistical submanifolds and a family of statistical connections on them. The geometric structures such as the second fundamental form, curvatures tensor, mean curvature, statistical Ricci curvature and the relations among them on a statistical submanifold of a statistical manifold equipped with $\mathcal{F}$-statistical connections are examined. The equations of Gauss and Codazzi of $\mathcal{F}$-statistical connections are obtained. Such structures when the statistical submanifolds are conjugate symmetric are discussed. We present a inequality for statistical submanifolds in real space forms with respect to $\mathcal{F}$-statistical connections. Also, we obtain a basic inequality involving statistical Ricci curvature and the squared $\mathcal{F}$-mean curvature of a statistical submanifold of statistical manifolds. Full article

This work investigates the effects on the factor manifolds of a singly warped product manifold resulting from the presence of a quasi-conformally flat, quasi-conformally symmetric, or divergence-free quasi-conformal curvature tensor. Quasi-conformally flat warped product manifolds exhibit three distinct scenarios: in one scenario, the base manifold has a constant curvature, while in the other two scenarios, it is quasi-Einstein. Alternatively, the fiber manifold has a constant curvature in two scenarios and is Einstein in one scenario. Quasi-conformally symmetric warped product manifolds present three distinct cases: in the first scenario, the base manifold is Ricci-symmetric and the fiber is Einstein; in the second case, the base manifold is Cartan-symmetric and the fiber has constant curvature; and in the last case, the fiber is Cartan-symmetric, and the Ricci tensor of the base manifold is of Codazzi type. Finally, conditions are provided for singly warped product manifolds that admit a divergence-free quasi-conformal curvature tensor to ensure that the Riemann curvature tensors of the factor manifolds are harmonic. Full article

In this paper, we focus on non-integrable distributions with a quarter-symmetric non-metric connection (QSNMC) in generalized Riemannian manifold. First, by studying a quarter-symmetric connection on the generalized Riemannian manifold, we obtain the condition that the connection is non-metric. Then, the Gauss, Codazzi and Ricci equations are proved for non-integrable distributions with respect to a quarter-symmetric non-metric connection in generalized Riemannian manifold. Furthermore, we deduce Chen’s inequalities for non-integrable distributions of real space forms with a quarter-symmetric non-metric connection in generalized Riemannian manifold as applications. After that, we give some examples of non-integrable distributions in Riemannian manifold with quarter-symmetric non-metric connection. Full article

We determine the general natural metrics G on the total space $TM$ of the tangent bundle of a Riemannian manifold $(M,g)$ such that the Schouten–van Kampen connection $\overline{\nabla }$ associated to the Levi-Civita connection of G is (quasi-)statistical. We prove that the base manifold must be a space form and in particular, when G is a natural diagonal metric, $(M,g)$ must be locally flat. We prove that there exist one family of natural diagonal metrics and two families of proper general natural metrics such that $(TM,\overline{\nabla },G)$ is a statistical manifold and one family of proper general natural metrics such that $(TM\setminus \left\{0\right\},\overline{\nabla },G)$ is a quasi-statistical manifold. Full article

In this paper, we address the study of the Kobayashi–Nomizu type and the Yano type connections on the tangent bundle $TM$ equipped with the Sasaki metric. Then, we determine the curvature tensors of these connections. Moreover, we find conditions under which these connections are torsion-free, Codazzi, and statistical structures, respectively, with respect to the Sasaki metric. Finally, we introduce the mutual curvature tensor on a manifold. We investigate some of its properties; furthermore, we study mutual curvature tensors on a manifold equipped with the Kobayashi–Nomizu type and the Yano type connections. Full article

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a Codazzi-type Ricci nearly Sasakian space form is either a Sasakian manifold with a constant $\varphi$-holomorphic sectional curvature $\mathcal{H}=1$ or a 5-dimensional proper nearly Sasakian manifold with a constant $\varphi$-holomorphic sectional curvature $\mathcal{H}>1$. We also prove that the spectrum of the operator $H^2$ generated by the nearly Sasakian space form is a set of a simple eigenvalue of 0 and an eigenvalue of multiplicity 4, and we induce that the underlying space form carries a Sasaki–Einstein structure. We show that there exist integrable distributions with totally geodesic leaves on the same manifolds, and we prove that there are no proper nearly Sasakian space forms with constant sectional curvature. Full article

The goal of the present research paper is to study how a spacetime manifold evolves when thermal flux, thermal energy density and thermal stress are involved; such spacetime is called a thermodynamical fluid spacetime (TFS). We deal with some geometrical characteristics of $TFS$ and obtain the value of cosmological constant $\mathsf{\Lambda }$. The next step is to demonstrate that a relativistic $TFS$ is a generalized Ricci recurrent $TFS$. Moreover, we use $TFS$ with thermodynamic matter tensors of Codazzi type and Ricci cyclic type. In addition, we discover the solitonic significance of $TFS$ in terms of the Ricci metric (i.e., Ricci soliton $RS$). Full article

The paper deals with the study of Z-symmetric manifolds ${\left(ZS\right)}_n$ admitting certain cases of Schouten tensor (specifically: Ricci-recurrent, cyclic parallel, Codazzi type and covariantly constant), and investigate some geometric and physical properties of the manifold. Moreover, we also study ${\left(ZS\right)}_4$ spacetimes admitting Codazzi type Schouten tensor. Finally, we construct an example of ${\left(ZS\right)}_4$ to verify our result. 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

Let ${\mathcal{M}}^k$ be a metric $\left(E^4=I\right)-$manifold equipped with electromagnetic-type structure E, a pseudo-Riemannian metric g and a nondegenerate $2-$form $\widehat{\omega }$. The paper deals with Codazzi couplings of an affine connection ∇ with E, g and $\widehat{\omega }$. We present some results concerning the relationship of these Codazzi couplings. In addition, we construct the connection between Codazzi couplings and $e-(E^4=I)$ Kaehler manifolds. Full article

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field $\mathbf{w}$ on a connected Riemannian manifold $(M,g)$ we show that for each non-constant smooth function $f\in C^{\infty }\left(M\right)$ there exists a non-zero vector field ${\mathbf{w}}^f$ associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold $(M,g)$ with an appropriate lower bound on the integral of the Ricci curvature $S({\mathbf{w}}^f,{\mathbf{w}}^f)$ gives a characterization of the odd-dimensional unit sphere ${\mathbf{S}}^{2m+1}$. Also, we show on an n-dimensional compact Riemannian manifold $(M,g)$ that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue $nc$ and the unit Killing vector field $\mathbf{w}$ satisfying ${∥\nabla \mathbf{w}∥}^2\le (n-1)c$ and Ricci curvature in the direction of the vector field $\nabla f-\mathbf{w}$ is bounded below by $\left(n-1\right)c$ is necessary and sufficient for $(M,g)$ to be isometric to the sphere ${\mathbf{S}}^{2m+1}\left(c\right)$. Finally, we show that the presence of a unit Killing vector field $\mathbf{w}$ on an n-dimensional Riemannian manifold $(M,g)$ with sectional curvatures of plane sections containing $\mathbf{w}$ equal to 1 forces dimension n to be odd and that the Riemannian manifold $(M,g)$ becomes a K-contact manifold. We also show that if in addition $(M,g)$ is complete and the Ricci operator satisfies Codazzi-type equation, then $(M,g)$ is an Einstein Sasakian manifold. Full article

In this work, the cases of non-integrable distributions in a Riemannian manifold with the first generalized semi-symmetric non-metric connection and the second generalized semi-symmetric non-metric connection are discussed. We obtain the Gauss, Codazzi, and Ricci equations in both cases. Moreover, Chen’s inequalities are also obtained in both cases. Some new examples based on non-integrable distributions in a Riemannian manifold with generalized semi-symmetric non-metric connections are proposed. Full article

In this study, we investigate generalized quasi-Einstein normal metric contact pair manifolds. Initially, we deal with the elementary properties and existence of generalized quasi-Einstein normal metric contact pair manifolds. Later, we explore the generalized quasi-constant curvature of normal metric contact pair manifolds. It is proved that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic parallel Ricci tensor and the Codazzi type of Ricci tensor are considered, and further prove that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we characterize normal metric contact pair manifolds satisfying certain curvature conditions related to $\mathcal{M}$-projective, conformal, and concircular curvature tensors. We show that a normal metric contact pair manifold with generalized quasi-constant curvature is locally isometric to the Hopf manifold $S^{2n+1}\left(1\right)\times S^1$. Full article