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 article, we obtain Chen inequalities on some submanifolds of complex space form. We obtain different results published by different geometer's as special cases.
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.
In this paper we count the number of $k$ -potent elements over $\mathbb{H}_{\mathbb{Z}_{p}}$ ,where $\mathbb{H}_{\mathbb{Z}_{p}}$ is the quaternion algebra over $\mathbb{Z}_{p}$ , and we present a descriptive formula for thegeneral case. For $k\in \{3,4,5\}$ , we give an explicit formula forthese values. Moreover, as an application of these results, we count thenumber of solutions of the equation $x^{k}=1$ over $\mathbb{H}_{\mathbb{Z}_{p}}$. For this purpose, we will use computer as a toolto check and understand the behavior of these elements in all cases that will be studied.
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 this paper, we study timelike translation surfaces with constant Gaussian curvature (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 3space and highlight a strong rigidity phenomenon in the timelike Lorentzian setting.
This note aims to study the cubic curve naturally associated to a triple of strictly positive numbers satisfying the Menelaus-Ceva condition. Restricting to a fixed triangle triangle we discuss several main points and the Lemoine line of triangle in this unified framework and we point out the cases when the cubic curve is an elliptic one.
In R2, let F be a fixed circle with centre C and radius r, and t a straight line at distance d of C. We study the curve which is the envelope of the circles whose centre lies on F and which are tangent to t. When d = 0 this curve is a nephroid, when d = 3r/2 it is a Cayley sextic.
In this work we obtained solutions of field equations for perfect fluid spherically symmetric static spacetimes and found their concircular vector fields (CCVFs) in f (R, T) gravity theory. It came out that special classes of these spacetimes possess either 4-dimensional or 15-dimensional CCVFs. In [28] the author obtained CCVFs for the same spacetimes in general relativity and it is argued that such spacetimes possess CCVFs of 4 , 5 , 6 and 15 dimensions. Our results revealed that the f (R, T) theory restricted the number of CCVFs for the same spacetime. f (R, T) theory allows such spacetimes to admit either the 4 basic Killing vector fields as CCVFs or compel the spacetime to be conformally flat and admit 15 CCVFs. We also calculated the energy density, fluid pressure, trace of the energy-momentum tensor T, Ricci scalar R and the function f (R, T). It is observed that energy density and fluid pressure of some solutions are related as p = rho which means that such particular metrics behave like dark energy models.
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.
In this paper, we prove the Chen-Ricci inequality for contact CR-warped products in the cosymplectic space forms, Theorem 5.1, which involves an intrinsic invariant (Ricci curvature) controlled by an extrinsic one (the mean curvature vector). This inequality is useful in both differential geometry and physics. In geometry, we apply it to get necessary conditions for the immersed submanifold to be minimal in a cosymplectic space form, which presents new answers for the well-known problem proposed by S.S. Chern, Problem 3. In physics, it enables us to derive some relations for the Dirichlet energy of the warping function controlled by some geometric invariants. In further research directions, we address a couple of open problems, namely Problem 4 and Problem 5, and a potential extension in the generalized contact metric manifolds.
In this paper, we give the characterizations of constant slope surfaces by means of the rotation minimizing frame (RMF). Also, constant slope surfaces are generalized to n-dimensional space En. Finally, some related examples are given.
In this paper, we introduce a new method for building ruled surfaces using dual quaternion curves and straight lines. We proved that the rigid body motion, arranged from any straight line in Euclidean 3-space R3 and any dual quaternion curve, produces a ruled surface in R3 when the parameter of the straight line is different from the parameter of the dual quaternion curve. We show that the ruled surface can be expressed by two separate rigid transformations. Moreover, we show that if the rotation part of the dual quaternion curve is a constant unit real quaternion, then the rigid body motion creates a developable ruled surface in R3. We show that this developable ruled surface also represents a generalized cylinder. Also, we give some examples to strengthen our results.
The aim of present paper is to study geometrical aspects of the almost Ricci soliton on D-homothetically deformed K-paracontact metric manifold and find the condition when the deformed metric remains almost Ricci soliton. Further, we analyze the nature of almost Ricci soliton on D-homothetically deformed K-paracontact metric manifold when associated potential vector field is divergence-free, the conformal vector field, or point wise collinear with Reeb vector field. We also provide examples of D-homothetically deformed K-paracontact metric manifold whose metric represents Ricci soliton. Finally, we discuss the behavior of almost gradient Ricci solitons on D-homothetically deformed K-paracontact metric manifold and obtain condition when the deformed metric remains the almost gradient Ricci soliton.
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).
In this paper, we investigate the geometric characterizations of 3-dimensional Lorentzian transSasakian manifolds admitting a generalized Z-tensor. First, we analyze the behavior of the generalized Z-tensor under certain symmetry conditions, such as Codazzi type, cyclic parallel, and phi-Z symmetric conditions, and derive the necessary constraints on the manifold structure. We also study the existence of generalized Z-Ricci solitons and prove that the existence of such a soliton implies that the scalar curvature is constant and the soliton generally exhibits a shrinking behavior.
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.
In this paper, we introduce and study the concept of f-osculating curves in both three and four dimensional Euclidean spaces (E3 and E4). 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 E3 and E4 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 article, we obtain Chen inequalities on some submanifolds of complex space form. We obtain different results published by different geometer's as special cases.
In this paper, we fully determine all Jacobi-type vector fields in hyperbolic three-space by taking advantage of both its constant negative curvature and its intrinsic compatibility with statistical manifold structures. The study is a natural extension of the results obtained by Wang and Zhang, [16], under the classical Levi-Civita connection.
