Searcharxiv⌕ Search

arXiv subjects

Fatma Almaz

Publications and source records attributed to Fatma Almaz.

12 recordsLinked to original sources
arXiv2026

Third-order relativistic kinematics and Siacci shock decompositions via localised Darboux frames

Constrained particle trajectories along lower-dimensional world sheets constitute a foundational domain in relativistic kinematics and submanifold theory. This paper establishes a comprehensive geometric and kinematic framework for third-order particle dynamics constrained to timelike surfaces in Minkowski 3-space. By utilising the localised Darboux frame, we explicitly derive the relativistically invariant parameters, namely the geodesic curvature, normal curvature, and geodesic torsion and analyse the non-linear coupling between the intrinsic surface topology and the extrinsic spacetime geometry. Primary focus is dedicated to formulating the relativistic "jerk" vector, which accounts for the instantaneous time rate of change of acceleration and captures the high-order structural stresses of the motion. Furthermore, we generalise Siacci's theorem to this Lorentzian setting, providing a non-perpendicular geometric decomposition of both the acceleration and jerk fields into explicit tangential and radial components relative to a fixed coordinate origin. Our formulations explicitly unveil the hidden central-force dynamics and dimensional compactification mechanisms under planar constraints, showing that out-of-plane torsional shocks vanish identically when the modified geodesic torsion is suppressed. These results offer novel analytical insights for structural stability and conserved angular momentum-like quantities governing constrained physical systems in non-Euclidean spacetimes.

math.GM↗
arXiv2026

A generalisation of g-rectifying and g-normal curves in Lorentzian n-space

In this paper, we introduce and analyze $g-$rectifying curves (spacelike and null curves) and $\ g-$normal curves in Lorentzian $n$-space, building upon the established notion of rectifying curves and normal curve, respectively. Our generalization extends this definition by considering an $% g-$position vector field, $\xi _{g}(s)=\int g(s)d\xi $, where $g$ is an integrable function in the arc-length parameter $s$. An $g$-rectifying curves(or $g-$normal curves) are then defined as an arc-length parametrized curve $\xi $ in Lorentzian $n-$space such that its $g$-position vector consistently lies within its rectifying space(or normal space). The primary objective of this work is to provide a comprehensive characterization and classification of these $g$-rectifying curves and $g-$normal curves, thereby expanding the geometric understanding of curves in Lorentzian $n$-spaces.

math.DG↗
arXiv2025

Smarandache curves and their properties on null curves in lightlike cone space $\mathbb{Q}_{2}^{3}$

This study investigates the differential geometric properties of Smarandache curves derived from null curves defined in the ligtlike cone space $% Q_{3}^{2}\subset E_{2}^{4}$. The indefinite metric structure causes the null vectors, and hence the null curves, to have a richer geometry in this space than in Euclidean or Minkowski spaces. In this study, we analyse the kinematic properties of the null curve using the null natural Frenet frame $% \{x,\xi ,N,W\}$. We then investigate the bending, torsion, and other geometric invariants of Smarandache curves constructed as linear combinations of these frame vectors (i.e., combinations of tangent, normal, or binormal vectors). The findings reveal how the original properties of the null curve are transferred to the Smarandache curves and how the metric of this particular space affects the characteristics of the Smarandache curves. This analysis provides a new perspective on the relationships between constrained and degenerate structures in differential geometry and light-like particle dynamics in theoretical physics.

math.GM↗
arXiv2025

On the invariant trajectories of lightlike paths near central potential singularities in semi-Riemannian manifolds of index 2

In this study, the geometric properties of null helices on a totally umbilical submanifold within a three-dimensional semi-Riemannian manifold of index 2 are investigated. The pseudo-Riemannian metric structure of semi-Riemannian manifold of index 2 and the fact that the submanifold is totally umbilical complicate the differential geometry of null helices. The study uses the given null Frenet frame to reveal the local properties of null helices (the curvatures $h,k_1,k_2$ of the null curves). By considering the degenerate metric condition of null helices due to the null tangent vector and the structure of the totally umbilical submanifold, equations and invariants characterizing null helices are obtained. Furthermore, this study establishes a novel structural rigidity theorem demonstrating that the simultaneous preservation of these degenerate curvature invariants under higher-order ambient parallel transport algebraically forces the collapsing of the mean curvature vector field ($H\equiv 0$), thereby identifying a clean boundary distinguishing minimal and totally geodesic embeddings in pseudo-Euclidean target spaces.

math.DG↗
arXiv2025

A review on the Vn-slant helices in lightlike cone Qn+1 En+2

In this paper, Vn-slant helices and the harmonic curvature functions of Vn-slant helices are de ned in lightlike cone Qn+1, and the differential equations of the harmonic curvature functions Hi, 1<i<n-2 of Vn-slant helices are expressed by using constant vector field W that is the axis of Vn slant helices. Also, the necessary and sufficient conditions are given according to the condition of being Vn-slant helices by using the asymptotic orthonormal frame in Qn+1.

math.DG↗
arXiv2025

Notes on the Geometry of Electromagnetic Fields and Maxwell's Equations along a non-null curves in non flat-3D space forms $M_{q}^{3}(c)$

In this paper, the directional derivatives in accordance with the orthonormal frame {T, N, B} are defined in $M_{q}^{3}(c)$, and the extended Serret-Frenet relations by using Frenet formulas are expressed. Furthermore, we express the bending elastic energy function for the same particle in $M_{q}^{3}(c)$ according to curve $\alpha (s,\xi ,\eta )$ and geometrical interpretation of the energy for unit vector fields and we also solve Maxwell's equations for the electric and magnetic field vectors in $M_{q}^{3}(c).$

math.DG↗
arXiv2025

A geometric perspective on the inextensible flows and energy of curves in 4-dimensional pseudo-Galilean space

In this study, inextensible flows of curves in four-dimensional pseudo-Galilean space are expressed, and the necessary and sufficient conditions of these curve flows are given as partial differential equations. Also, the directional derivatives are defined in accordance with the Serret-Frenet frame in G41, the extended Serret-Frenet relations are expressed by using Frenet formulas in G41. Furthermore, the bending elastic energy functions are expressed for the same particle according to curve a(s,t).

math.DG↗
arXiv2025

The investigation of some special curves in the pseudo-Galilean 4-space $G_{1}^{4} $

In this paper, we investigate and characterise an arbitrary admissible curve in terms of its curvature functions in the pseudo-Galilean space $G_{1}^{4}$% . We also give some special curves in four-dimensional pseudo-Galilean space and their characterisations, such as position curve, osculating curve, normal curve, rectifying curve, slant helices, and spherical curves. Moreover, this study provides classifications of admissible curves in $G_{1}^{4}$. That is, the admissible rectifying curves lying fully in the $G_{1}^{4}$ are expressed and given necessary and sufficient conditions for such a curve to be an admissible rectifying curve, osculating curve, normal curve, slant helices and spherical curves in $G_{1}^{4}$.

math.DG↗
arXiv2023

The Research on Rotational Surfaces in pseudo Euclidean 4-space with index 2

In this study, we define a brief description of the hyperbolic and elliptic rotational surfaces using a curve and matrices in 4-dimensional semi Euclidean space. That is, we provide different types of rotational matrices, which are the subgroups of M by rotating a selected axis in E4. Hence, we choose two parameter matrices groups of rotations and we give the matrices of rotation corresponding to the appropriate subgroup in 4-dimensional semi Euclidean space and we generate rotated surfaces.

math.DG↗
arXiv2021

The physical approach on the surfaces of rotation in E24

In this paper, some physical expressions as the specific energy and the specific angular momentum on these surfaces of rotation are investigated with the help of Clairaut's theorem using the conditions being geodesic in which the curves can be chosen to be time-like curves, which allows us to constitute the specific energy and specific angular momentum

math.DG↗
arXiv2017

Assesment of Smarandache Curves in the Null Cone Q^2

In this paper, we give Smarandache curves according to the asymptotic orthonormal frame in null cone Q^2. By using cone frame formulas, we present some characterizations of Smarandache curves and calculate cone frenet invariants of these curves. Also, we illustrate these curves with an example.

math.GM↗