Ruled surfaces generated by elliptic cylindrical curves in the isotropic space

2019 ◽  
Vol 26 (3) ◽  
pp. 331-340 ◽  
Author(s):  
Muhittin Evren Aydin ◽  
Adela Mihai
Keyword(s):  
Vector Field ◽  
Constant Curvature ◽  
Ruled Surfaces ◽  
Laplacian Operator ◽  
Position Vector ◽  
Isotropic Space

Abstract In this paper we study the ruled surfaces generated by elliptic cylindrical curves in the isotropic 3-space {\mathbb{I}^{3}} . We classify such surfaces in {\mathbb{I}^{3}} with constant curvature and satisfying an equation in terms of the components of the position vector field and the Laplacian operator. Several examples are given and illustrated by figures.

scholarly journals Classification of Some Special Types Ruled Surfaces in Simply Isotropic 3-Space

2017 ◽  
Vol 55 (1) ◽  
pp. 87-98 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Nural Yuksel
Keyword(s):  
Real Number ◽  
Laplace Operator ◽  
Fundamental Form ◽  
Three Dimensional ◽  
Ruled Surfaces ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
The Laplace Operator ◽  
First Fundamental Form

AbstractIn this paper, we classify two types ruled surfaces in the three dimensional simply isotropic space I13under the condition ∆xi= λixiwhere ∆ is the Laplace operator with respect to the first fundamental form and λ is a real number. We also give explicit forms of these surfaces.

scholarly journals Classification of generalized Yamabe solitons in Euclidean spaces

2021 ◽  
pp. 2150022
Author(s):  
Shunya Fujii ◽  
Shun Maeta
Keyword(s):  
Vector Field ◽  
Position Vector ◽  
Euclidean Spaces ◽  
Yamabe Solitons ◽  
Gradient Solitons

In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, [Formula: see text]-almost Yamabe solitons, gradient [Formula: see text]-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.

A link between torse-forming vector fields and rotational hypersurfaces

2017 ◽  
Vol 14 (12) ◽  
pp. 1750177 ◽  
Author(s):  
Bang-Yen Chen ◽  
Leopold Verstraelen
Keyword(s):  
Vector Field ◽  
Riemannian Space ◽  
Vector Fields ◽  
Natural Extension ◽  
Mathematical Physics ◽  
Tangential Component ◽  
Position Vector ◽  
Axis Of Rotation

Torse-forming vector fields introduced by Yano [On torse forming direction in a Riemannian space, Proc. Imp. Acad. Tokyo 20 (1944) 340–346] are natural extension of concurrent and concircular vector fields. Such vector fields have many nice applications to geometry and mathematical physics. In this paper, we establish a link between rotational hypersurfaces and torse-forming vector fields. More precisely, our main result states that, for a hypersurface [Formula: see text] of [Formula: see text] with [Formula: see text], the tangential component [Formula: see text] of the position vector field of [Formula: see text] is a proper torse-forming vector field on [Formula: see text] if and only if [Formula: see text] is contained in a rotational hypersurface whose axis of rotation contains the origin.

scholarly journals An integral formula for compact hypersurfaces in a Euclidean space and its applications

1992 ◽  
Vol 34 (3) ◽  
pp. 309-311 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Smooth Function ◽  
Support Function ◽  
Integral Formula ◽  
Position Vector ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field

Let M be a compact hypersurface in a Euclidena space ℝn+1. The support function p of M is the component of the position vector field of Min ℝn+1 along the unit normal vector field to M, which is a smooth function defined on M. Let S be the scalar curvature of M. The object of the present paper is to prove the following theorems.

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals Constant curvature surfaces in a pseudo-isotropic space

2018 ◽  
Vol 49 (3) ◽  
pp. 221-233 ◽  
Author(s):  
Muhittin Evren Aydin
Keyword(s):  
Constant Curvature ◽  
Local Structure ◽  
Mean Curvature ◽  
Plane Curve ◽  
Surfaces Of Revolution ◽  
Curves And Surfaces ◽  
Isotropic Space ◽  
Constant Curvature Surfaces

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space $\mathbb{I}_{p}^{3}$ that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron for spacelike and timelike curves, respectively. The causal character of all admissible surfaces in $\mathbb{I}_{p}^{3}$ has to be timelike up to its absolute. We introduce the formulas of Gaussian and mean curvature for timelike surfaces in $\mathbb{I}_{p}^{3}$. As applications, we describe the surfaces of revolution which are the orbits of a plane curve under a hyperbolic rotation with constant Gaussian and mean curvature.

On Backlund transformation and motion of null Cartan curves

2021 ◽  
Author(s):  
Melek Erdoğdu ◽  
Ayşe Yavuz
Keyword(s):  
Angular Momentum ◽  
Constant Curvature ◽  
Vector Fields ◽  
Bäcklund Transformation ◽  
Position Vector ◽  
Angular Momentum Vector ◽  
Backlund Transformation ◽  
Momentum Vector ◽  
Bertrand Curve ◽  
Modern Physics

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector [Formula: see text] In the case of [Formula: see text] we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve [Formula: see text]

Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature

2017 ◽  
Vol 69 (5) ◽  
pp. 961-991 ◽  
Author(s):  
Jaime Andrade ◽  
Nestor Dàvila ◽  
Ernesto Pérez-Chavela ◽  
Claudio Vidal
Keyword(s):  
Angular Momentum ◽  
Vector Field ◽  
Constant Curvature ◽  
Kepler Problem ◽  
Phase Portraits ◽  
Problem Of Time

AbstractWe classify and analyze the orbits of the Kepler problemon surfaces of constant curvature (both positive and negative, 𝕊2and ℍ2, respectively) as functions of the angular momentum and the energy. Hill's regions are characterized, and the problem of time-collision is studied. We also regularize the problem in Cartesian and intrinsic coordinates, depending on the constant angular momentum, and we describe the orbits of the regularized vector field. The phase portraits both for 𝕊2and ℍ2are pointed out.

scholarly journals Ruled Surfaces of Generalized Self-Similar Solutions of the Mean Curvature Flow

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Rafael López
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Position Vector ◽  
Translation Surfaces ◽  
Self Similar ◽  
The Mean ◽  
Similar Solutions

AbstractIn Euclidean space, we investigate surfaces whose mean curvature H satisfies the equation $$H=\alpha \langle N,{\mathbf {x}}\rangle +\lambda $$ H = α ⟨ N , x ⟩ + λ , where N is the Gauss map, $${\mathbf {x}}$$ x is the position vector, and $$\alpha $$ α and $$\lambda $$ λ are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.

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