In a 3-dimensional generalized trans-Sasakian manifold, explicit formula for Ricci operator, Ricci tensor and curvature tensor are obtained. In particular, expressions for Ricci tensor are obtained in a 3-dimensional generalized trans-Sasakian manifold in cases of the manifold being quasi-Einstein or generalized quasi-Einstein.

In $\mathbb{R}^2$, let $\Gamma$ be a fixed circle with centre $C$ and radius $r$,and $\ell$ a straight line at distance $d$ of $C$. We study the curve which is the envelope of the circles whose centre lies on $\Gamma$ and which are tangent to $\ell$. When $d=0$ this curve is a nephroid, when $d=3r/2$ it is a Cayley sextic.

In this paper, we have given the characterization of constant slope surfaces with the by of the rotation minimizing frame (RMF). Also, constant slope surfaces are generalized n-dimensional space $E^n$. Finally, a related example is given.

The sectional curvature, Ricci curvature, and scalar curvature for a product generalized Sasakian space form are obtained. Furthermore, the Chen-Ricci inequality and the Hineva inequality are established for submanifolds of a product generalized Sasakian space form, including product Sasakian, product cosymplectic, and product Kenmotsu space forms. The equality cases are also discussed.

This paper considers the problem of integral geometry in three-dimensional space over a family of cones with a weight function of a special form. A uniqueness theorem for the solution to the Fourier image is proved and two examples are considered to obtain a more accurate solution to the problem.

In this work, we present an adaptation of the pole-based central projection from the sphere to the ellipsoid and the elliptic paraboloid. We begin by constructing the central pole-to-plane projections for each quadric surface separately, analyzing their geometric particularities and the challenges arising from variable curvatures and, in the case of the paraboloid, non-compactness. A key geometric insight reveals that the projected ellipses on the $xy$-plane and the corresponding conic sections on the quadrics are related by a homothety. This fundamental relationship allows us to establish unified scaling laws for their geometric invariants: the curvature scales by $\lambda^{-1}$, the arc length by $\lambda$, and the area by $\lambda^2$, where $\lambda$ is the homothety factor. These results provide a complete characterization of the eccentricities, curvatures, arc lengths, and areas of the intersecting conics and their projections.

This paper introduces and investigates a generalization of the notion of a pointed Riemannian manifold having its radial curvature bounded from above by that of a model surface of revolution.

In this paper, we introduce and study the concept of $f$-osculating curves in both three and four dimensional Euclidean spaces ($\mathbb{E}^{3}$ and $\mathbb{E}^{4}$). These curves are characterized by the condition that their $f$-position vectors lie in the osculating plane. We establish necessary and sufficient conditions for a unit-speed curve in $\mathbb{E}^3$ and $\mathbb{E}^4$ to be an $f$-osculating curve and derive relations among their curvature functions. We also examine special cases in which one or more curvature functions are constant, analyzing the behaviour of the remaining curvature functions.

In this paper, we study timelike translation surfaces with constant Gaussian curvature \textbf{(CGC)} in the three-dimensional Minkowski space. Such surfaces are generated as the sum of two timelike space curves and naturally arise in the context of Lorentzian surface geometry. By employing a detailed analytic and geometric approach, we prove that any timelike translation surface with constant Gaussian curvature must be flat. As a consequence, we show that the only timelike translation surfaces satisfying this curvature condition are cylindrical surfaces. Furthermore, we establish that timelike translation surfaces with constant Gaussian curvature cannot be minimal everywhere. As a geometric characterization of the generating curves, we prove that one of the curves must necessarily be either a timelike hyperbola or a straight line. These results provide a complete local classification of timelike translation surfaces with constant Gaussian curvature in Minkowski 3-space and highlight a strong rigidity phenomenon in the timelike Lorentzian setting.

In this paper, we present a characterization of the complement of the set of points of a hyperbolic quadric of PG(3, q). As a byproduct we obtain a generalization of a recent result of B. Sahu [A characterisation of the planes meeting a hyperbolic quadric of PG(3, q) in a conic, Austral. J. Combin. 84 (1), (2022) 178-186] characterizing the set of non tangent planes to a hyperbolic quadric of PG(3, q).