Search Results (10)

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

This paper explores the geometry of 3-dimensional quasi Sasakian manifolds under CL-transformations. We construct both infinitesimal and $CL$-transformation and demonstrate that the former does not necessarily yield projective killing vector fields. A novel invariant tensor, termed the $CL$-curvature tensor, is introduced and shown to remain invariant under $CL$-transformations. Utilizing this tensor, we characterize $CL$-flat, $CL$-symmetric, $CL$-$\phi$ symmetric and $CL$-$\phi$ recurrent structures on such manifolds by mean of differential equations. Furthermore, we investigate conditions under which a Ricci soliton exists on a CL-transformed quasi Sasakian manifold, revealing that under flat curvature, the structure becomes Einstein. These findings contribute to the understanding of curvature dynamics and soliton theory within the context of contact metric geometry. Full article

In this study, we investigated the geometric and physical implications of the $W_3$ curvature tensor within the framework of $f(R,G)$ gravity. We found the sufficient conditions for $W_3$ flat spacetimes with constant scalar curvature to be de Sitter ($R>0$) or Anti-de Sitter ($R<0$) models. The properties of isotropic spacetime in the modified gravity framework were also investigated. Furthermore, we explored spacetimes with a divergence-free $W_3$ curvature tensor. The necessary and sufficient condition for a $W_3$ Ricci recurrent and parallel spacetime to transform into an Einstein spacetime was determined. Finally, we analyzed the role of the $W_3$ curvature tensor in black hole thermodynamics within $f(R,G)$ gravity. Full article

In the present article, we classify conformally flat weakly Ricci-symmetric space-times and obtain that they represent Robertson–Walker space-times. Furthermore, we provethat a Ricci-recurrent weakly Ricci-symmetric space-time is static and a Ricci-semi-symmetric weakly Ricci-symmetric space-time does not exist. Further, we acquire the conditions under which a weakly Ricci-symmetric twisted space-time becomes a generalized Robertson–Walker space-time. Also, we examine the effect of conformally flat weakly Ricci-symmetric space-time solutions in $f(R,G)$ gravity by considering two models, and we see that the null, weak and strong energy conditions are verified, but the dominant energy condition fails, which is also consistent with present observational studies that reveal the universe is expanding. Finally, we apply the flat Friedmann–Robertson–Walker metric to deduce a relation between deceleration, jerk and snap parameters. Full article

Dynamic network representation learning has recently attracted increasing attention because real-world networks evolve over time, that is nodes and edges join or leave the networks over time. Different from static networks, the representation learning of dynamic networks should not only consider how to capture the structural information of network snapshots, but also consider how to capture the temporal dynamic information of network structure evolution from the network snapshot sequence. From the existing work on dynamic network representation, there are two main problems: (1) A significant number of methods target dynamic networks, which only allow nodes to increase over time, not decrease, which reduces the applicability of such methods to real-world networks. (2) At present, most network-embedding methods, especially dynamic network representation learning approaches, use Euclidean embedding space. However, the network itself is geometrically non-Euclidean, which leads to geometric inconsistencies between the embedded space and the underlying space of the network, which can affect the performance of the model. In order to solve the above two problems, we propose a geometry-based dynamic network learning framework, namely DyLFG. Our proposed framework targets dynamic networks, which allow nodes and edges to join or exit the network over time. In order to extract the structural information of network snapshots, we designed a new hyperbolic geometry processing layer, which is different from the previous literature. In order to deal with the temporal dynamics of the network snapshot sequence, we propose a gated recurrent unit (GRU) module based on Ricci curvature, that is the RGRU. In the proposed framework, we used a temporal attention layer and the RGRU to evolve the neural network weight matrix to capture temporal dynamics in the network snapshot sequence. The experimental results showed that our model outperformed the baseline approaches on the baseline datasets. Full article

The purpose of this study is to evaluate the curvature tensor and the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection on the tangent bundle $TM$. Certain results on a semisymmetric P-Sasakian manifold, generalized recurrent P-Sasakian manifolds, and pseudo-symmetric P-Sasakian manifolds on $TM$ are proved. Full article

A conformally flat GRW space-time is a perfect fluid RW space-time. In this note, we investigated the influence of many differential curvature conditions, such as the existence of recurrent and semi-symmetric curvature tensors. In each case, the form of the Ricci curvature tensor, the energy–momentum tensor, the energy density, the pressure of the fluid, and the equation of state are determined and interpreted. For example, it is demonstrated that a Ricci semi-symmetric RW space-time reduces to Einstein space-time or a Ricci recurrent RW space-time, and the perfect fluid space-time is referred to as Yang pure space-time or dark matter era. 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 geodesic mappings of spaces with affine connections onto generalized 2-, 3-, and m-Ricci-symmetric spaces. In either case, the main equations for the mappings are obtained as a closed system of linear differential equations of the Cauchy type in the covariant derivatives. For the systems, we have found the maximum number of essential parameters on which the solutions depend. These results generalize the properties of geodesic mappings onto symmetric, recurrent, and also 2-, 3-, and m-(Ricci-)symmetric spaces with affine connections. Full article