arXiv subjects

Muhittin Evren Aydin

Publications and source records attributed to Muhittin Evren Aydin.

At least 19 recordsLinked to original sources

arXiv2026

Curves in Riemannian Manifolds Making Prescribed Angles With Torse-Forming Vector Fields

In this paper, we introduce the notion of a prescribed angle curve in a Riemannian manifold associated with a pair $(\mathcal{V},θ)$, where $\mathcal{V}$ is a unit vector field along the curve and $θ$ denotes the angle between $\mathcal{V}$ and the principal normal vector of the curve. When $\mathcal{V}$ is a torse-forming vector field, we establish an existence result for prescribed angle curves. In the $3$-dimensional case, we determine the curvatures of these curves in terms of the prescribed angle and the potential function of $\mathcal{V}$. Moreover, using this notion, we provide a new characterization of curves lying on geodesic spheres in real space forms.

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arXiv2026

Generalization of the catenary in the dual plane

In this paper, we study a dual analogue of the classical catenary within the class of admissible curves in the dual plane $\mathbb{D}^2$. We introduce $α$-catenaries in $\mathbb{D}^2$ as stationary points of a potential energy functional, where $α\in \mathbb{R}$ is a real parameter. We derive the corresponding Euler-Lagrange equations and obtain explicit equations of these curves for specific values of $α$. Furthermore, we establish a geometric characterization of $α$-catenaries in terms of their curvature and unit normal vector field.

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arXiv2025

Extension of a problem of Euler in $\mathbb{H}^2$ and in $\mathbb{S}^2$

In this paper, we extend the notion of stationary curves with respect to the moment of inertia from a point $N$ in the Euclidean plane $\mathbb{R}^2$ to the case that the ambient space is either the hyperbolic plane $\mathbb{H}^2$ or the sphere $\mathbb{S}^2$. We characterize the critical points of this energy in terms of the curvature of the curve and the distance to $N$. In $\mathbb{H}^2$, we prove that the only closed stationary curves are circles centered at $N$. In $\mathbb{S}^2$, we estimate the value of $α$ for closed curves according to the hemisphere of $\mathbb{S}^2$ in which the curve lies. In addition, we find the first integrals of the ODEs that describe the parametrizations of stationary curves in both ambient spaces. Finally, we consider the energy minimization problem for curves connecting two points collinear with $N$, in particular solving the case of geodesics.

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arXiv2025

Extension of a problem of Euler in Lorentz-Minkowski plane

In this paper we study curves in Lorentz-Minkowski space $\mathbb{L}^2$ that are critical points of the moment of inertia with respect to the origin. This extends a problem posed by Euler in the Lorentzian setting. We obtain explicit solutions for stationary curves in $\mathbb{L}^2$ distinguishing if the curve is spacelike or timelike. We also give a method to carry stationary spacelike curves into stationary timelike curves and vice versa via symmetries and inversions about the lightlike cone. Finally, we solve the problem of maximizing the energy among all spacelike curves joining two given points which are collinear with the origin.

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arXiv2025

Classification of ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia

In this paper, we study hypersurfaces in Lorentz-Minkowski space $\mathbb{L}^{n+1}$ that are stationary for the moment of inertia with respect to the origin. After giving examples and applications of the maximum principle, we classify, in $\mathbb{L}^3$, all stationary surfaces that are ruled surfaces. We prove that planes are the only cylindrical stationary surfaces. If the surface is not cylindrical, the classification depends on the causal character of the rulings. In addition, we provide explicit parametrizations of all ruled stationary surfaces.

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arXiv2025

Invariant $λ$-translators for the Gauss curvature flow in Euclidean space

A $λ$-translator is a surface in Euclidean space $\mathbb{R}^3$ whose Gauss curvature $K$ satisfies $K=\langle N, \vec{v} \rangle +λ$, where $N$ is the Gauss map, $\vec{v}$ is a fixed direction, and $λ\in \mathbb{R}$. In this paper, we classify all $λ$-translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.

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arXiv2025

Torqued and Anti-Torqued Vector Fields on Hyperbolic Spaces

In this paper, we study the existence of torqued and anti-torqued vector fields on the hyperbolic ambient space $\mathbb{H}^n$. Although there are examples of proper torqued vector fields on open subsets of $\mathbb{H}^n$, we prove that there is not a proper torqued vector field globally defined on $\mathbb{H}^n$. Similarly, we have another non-existence result for anti-torqued vector fields as long as their conformal scalar is a non-constant function satisfying certain conditions. When the conformal scalar is constant, some examples of anti-torqued vector fields are provided.

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arXiv2025

Anti-torqued slant helices and Torqued Curves in Riemannian manifolds

In this paper, we introduce the notion of an anti-torqued slant helix in a Riemannian manifold, defined as a curve whose principal vector field makes a constant angle with an anti-torqued vector field globally defined on the ambient manifold. We characterize and classify such curves through systems of differential equations involving their invariants. Several illustrative examples are also provided. Finally, we study torqued curves, defined as curves for which the inner product function of the principal vector field and a torqued vector field along the curve is a given constant.

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arXiv2024

Classification of translation surfaces in $\mathbb{R}^3$ with constant sectional curvature

In this paper, we study translation surfaces in the Euclidean space endowed with a canonical semi-symmetric non-metric connection. We completely classify the translation surfaces of constant sectional curvature with respect to this connection, proving that they are generalized cylinders. This consequence is the same as in the case of the Levi-Civita connection, but in this new setting, there are also generalized cylinders whose sectional curvature can be constant or non-constant, in contrast to the Levi-Civita connection, where the Gaussian curvature is zero.

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arXiv2024

Constant sectional curvature surfaces with a semi-symmetric non-metric connection

Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to this connection is constant. In case that the surface is cylindrical, we obtain full classification when the rulings are orthogonal or parallel to $\mathsf{C}$. If the surface is rotational, we prove that the rotation axis is parallel to $\mathsf{C}$ and we classify all conical rotational surfaces with constant sectional curvature. Finally, for the particular case $\frac12$ of the sectional curvature, the existence of rotational surfaces orthogonally intersecting the rotation axis is also obtained.

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arXiv2024

Rectifying submanifolds of Riemannian manifolds with anti-torqued axis

In this paper we study rectifying submanifolds of a Riemannian manifold endowed with an anti-torqued vector field. For this, we first determine a necessary and sufficient condition for the ambient space to admit such a vector field. Then we characterize submanifolds for which an anti-torqued vector field is always assumed to be tangent or normal. A similar characterization is also done in the case of the torqued vector fields. Finally, we obtain that the rectifying submanifolds with anti-torqued axis are the warped products whose warping function is a first integration of the conformal scalar of the axis.

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arXiv2023

Classification of separable hypersurfaces with constant sectional curvature

In this paper, we give a full classification of the separable hypersurfaces of constant sectional curvature in the Euclidean $n$-space $\mathbb{R}^n$. In dimension $n=3$, this classification was solved by Hasanis and López [Manuscripta Math. 166, 403-417 (2021)]. When $n>3$, we prove that the separable hypersurfaces of null sectional curvature are three particular families of such hypersurfaces. Finally, we prove that hyperspheres are the only separable hypersurfaces with nonzero constant sectional curvature.

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arXiv2023

Singular Miminal Ruled Surfaces

In this paper we study surfaces with minimal potential energy under gravitational forces, called singular minimal surfaces. We prove that a singular minimal ruled surface in a Euclidean $3-$space is cylindrical, in particular as an $α-$catenary cylinder by a result of López [Ann. Glob. Anal. Geom. 53(4) (2018), 521-541]. This result is also extended in Lorentz-Minkowski $3-$space.

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arXiv2023

A non-Newtonian approach in differential geometry of curves: multiplicative rectifying curves

In this paper, we study the rectifying curves in multiplicative Euclidean space of dimension 3, i.e., those curves for which the position vector always lies in its rectifying plane. Since the definition of rectifying curve is affine and not metric, we are directly able to perform multiplicative differential-geometric concepts to investigate such curves. Having presented several characterizations, we completely classify the multiplicative rectifying curves by means of the multiplicative spherical curves.

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arXiv2022

Effect of local fractional derivatives on Riemann curvature tensor

In this short note, we investigate the effect of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated $M-$ and $\mathcal{V}-$fractional derivatives. Then, according to this general operator, a particular Riemannian metric tensor field on the real affine space $\mathbb{R}^{n}$ that is different than Euclidean one is defined. In conlusion, we obtain that the Riemann curvature tensor of $\mathbb{R}^{n}$ endowed with this particular metric is identically $0$, namely, locally isometric to Euclidean space.

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arXiv2022

Rotational $K^α$-translators in Minkowski space

A spacelike surface in Minkowski space $\mathbb{R}_1^3$ is called a $K^α$-translator of the flow by the powers of Gauss curvature if satisfies $K^α= \langle N,\vec{v}\rangle$, $α\neq 0$, where $K$ is the Gauss curvature, $N$ is the unit normal vector field and $\vec{v}$ is a direction of $\mathbb{R}_1^3$. In this paper, we classify all rotational $K^α$-translators. This classification will depend on the causal character of the rotation axis. Although the theory of the $K^α$-flow holds for spacelike surfaces, the equation describing $K^α$-translators is still valid for timelike surfaces. So we also investigate the timelike rotational surfaces that satisfy the same prescribing Gauss curvature equation.

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arXiv2022

Translators of the Gauss curvature flow

A $K^α$-translator is a surface in Euclidean space $\r^3$ that moves by translations in a spatial direction and under the $K^α$-flow, where $K$ is the Gauss curvature and $α$ is a constant. We classify all $K^α$-translators that are rotationally symmetric. In particular, we prove that for each $α$ there is a $K^α$-translator intersecting orthogonally the rotation axis. We also describe all $K^α$-translators invariant by a uniparametric group of helicoidal motions and the translators obtained by separation of variables.

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arXiv2021

Translating solitons of translation and homothetical types

We prove that if a translating soliton can be expressed as the sum of two curves and one of these curves is planar, then the other curve is also planar and consequently the surface must be a plane or a grim reaper. We also investigate translating solitons that can be locally written as the product of two functions of one variable. We extend the results in Lorentz-Minkowski space.

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