We show that a strictly almost cosympletic $3$-manifold $M$ is weakly $\eta$-Einstein if and only if $M$ is weakly Einstein. On the other hand, a cosympletic $3$-manifold $M$ is weakly $\eta$-Einstein if and only if $M$ is $\eta$-Einstein. If the scalar curvature $r$ of a cosymplectic $3$-manifold $M$ is zero, then $M$ is Einstein, weakly Einstein, and flat.
The polarization dynamics of light in optical fibers are strongly influenced by the geometry of the propagation path. While experimental optics often employs Jones matrices or Mueller calculus to describe polarization transformations, such formalisms do not explicitly capture the geometric and topological features of curved trajectories. In this work, we propose a purely geometric framework by modelling an optical fiber as a framed curve in three-dimensional space. Using Berry phase analysis and Fermi–Walker transport, we establish explicit relations between curvature–torsion invariants of the curve and the evolution of the polarization vector. This approach leads to the definition of electromagnetic curves, providing a consistent mathematical formalism to interpret polarization transport in structured fibers. Theoretical results are illustrated through case studies and symbolic computations in MAPLE, which demonstrate how singularities in the framed curve influence the polarization state. Although our study is primarily mathematical, it offers a complementary theoretical perspective to standard optical methods and highlights how geometric invariants can serve as tools for understanding polarization in non-planar fiber geometries.
In this paper, we study quadrilaterals $ABCD$ in the isotropic plane that are determined by the vertex angles $\alpha=\angle(DA,AB)$, $\beta=\angle(AB,BC)$, $\gamma=\angle(BC,CD)$ and a diagonal angle $\varepsilon=\angle(AC,BD)$. We obtain the loci of diagonal points and the envelope of the sides when only some of the angles are given.
This paper investigates $3$-dimensional generalized trans-Sasakian manifolds admitting generalized $\eta$-Ricci solitons. We establish necessary and sufficient conditions for the existence of such solitons and provide a complete characterization in the three-dimensional case. The obtained results clarify the interplay between the soliton equation and the underlying almost contact metric manifold. Explicit and concrete examples are presented to illustrate these results.
We introduce a novel horizontally deformed Cheeger--Gromoll metric on the tangent bundle of a Riemannian manifold, thereby enlarging the spectrum of invariant metrics available in this context. A systematic analysis is performed on the geometric characteristics engendered by this metric, yielding explicit expressions for the Levi--Civita connection and the full array of curvature tensors. Moreover, we examined the geometry of a hypersurface embedded in the tangent bundle, providing a detailed analysis of its intrinsic structure, induced Levi--Civita connection, and associated curvature tensors.
