scholarly journals Rectifying curves and geodesics on a cone in the Euclidean 3-space

2017 ◽  
Vol 48 (2) ◽  
pp. 209-214 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Vector Field ◽  
Position Vector ◽  
Arbitrary Cone ◽  
Rectifying Curve

A twisted curve in the Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lie in its rectifying plane. In this article we study geodesics on an arbitrary cone in $\mathbb E^3$, not necessary a circular one, via rectifying curves. Our main result states that a curve on a cone in $\mathbb E^3$ is a geodesic if and only if it is either a rectifying curve or an open portion of a ruling. As an application we show that the only planar geodesics in a cone in $\mathbb E^3$ are portions of rulings.

scholarly journals On f-rectifying curves in the Euclidean 4-space

2021 ◽  
Vol 13 (1) ◽  
pp. 192-208
Author(s):  
Zafar Iqbal ◽  
Joydeep Sengupta
Keyword(s):  
Vector Field ◽  
Integrable Function ◽  
Orthogonal Complement ◽  
Position Vector ◽  
Normal Vector ◽  
Arc Length ◽  
Normal Vector Field ◽  
Parametrized Curve ◽  
Rectifying Curve

Abstract A rectifying curve in the Euclidean 4-space 𝔼4 is defined as an arc length parametrized curve γ in 𝔼4 such that its position vector always lies in its rectifying space (i.e., the orthogonal complement Nγ ˔ of its principal normal vector field Nγ) in 𝔼4. In this paper, we introduce the notion of an f-rectifying curve in 𝔼4 as a curve γ in 𝔼4 parametrized by its arc length s such that its f-position vector γf, defined by γf (s) = ∫ f(s)dγ for all s, always lies in its rectifying space in 𝔼4, where f is a nowhere vanishing integrable function in parameter s of the curve γ. Also, we characterize and classify such curves in 𝔼4.

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.

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 Affine focal sets of codimension-2 submanifolds contained in hypersurfaces

2018 ◽  
Vol 148 (5) ◽  
pp. 995-1016
Author(s):  
Marcos Craizer ◽  
Marcelo J. Saia ◽  
Luis F. Sánchez
Keyword(s):  
Vector Field ◽  
Single Line ◽  
Position Vector ◽  
General Description ◽  
Affine Sphere ◽  
Affine Normal ◽  
Darboux Vector ◽  
Focal Set ◽  
Regular Hypersurface ◽  
Affine Metric

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.

Space—times admitting Killing—Yano tensors. II

1982 ◽  
Vol 381 (1781) ◽  
pp. 315-322 ◽  
Keyword(s):  
Angular Momentum ◽  
Vector Field ◽  
Equatorial Plane ◽  
Kepler Problem ◽  
Three Dimensional ◽  
Position Vector ◽  
Line Elements ◽  
Tangent Field

In this paper we derive canonical line elements admitting a simple KillingYano tensor f* ab. There exist three distinct cases according to the character o f f* ab (spacelike, timelike, null). We reveal several close analogies between the vector field la = f * arp r along a geodesic with tangent field p r and the angular momentum l = rxpin the case of a spacelike Killing-Yano tensor. In particular, we show that, in consequence of the Killing-Yano tensor equations, there exists an analogue of the three-dimensional position vector field in certain hypersurfaces and th at la can be written in the form (r x p )a. Furthermore, an analogue of the ‘ equatorial plane ’ of the classical Kepler problem can be constructed intrinsically.

scholarly journals A new aspect of rectifying curves and ruled surfaces in Galilean 3-space

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2953-2962
Author(s):  
Çetin Demir ◽  
İsmail Gök ◽  
Yusuf Yayli
Keyword(s):  
Vector Fields ◽  
The Other ◽  
Ruled Surfaces ◽  
Position Vector ◽  
Darboux Vector ◽  
Minkowski Spaces ◽  
Base Curve ◽  
Rectifying Curve

A curve is named as rectifying curve if its position vector always lies in its rectifying plane. There are lots of papers about rectifying curves in Euclidean and Minkowski spaces. In this paper, we give some relations between extended rectifying curves and their modified Darboux vector fields in Galilean 3-Space. The other aim of the paper is to introduce the ruled surfaces whose base curve is rectifying curve. Further, we prove that the parameter curve of the surface is a geodesic.

scholarly journals Characterization of Rectifying Curves by Their Involutes and Evolutes

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3077
Author(s):  
Marilena Jianu ◽  
Sever Achimescu ◽  
Leonard Dăuş ◽  
Adela Mihai ◽  
Olimpia-Alice Roman ◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Position Vector ◽  
Arc Length ◽  
Remarkable Property ◽  
Spherical Curve ◽  
Pass Through ◽  
Cartesian Coordinate ◽  
Necessary And Sufficient ◽  
Rectifying Curve

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

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