scholarly journals Slant Curves in the Unit Tangent Bundles of Surfaces

ISRN Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Zhong Hua Hou ◽  
Lei Sun
Keyword(s):  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sasaki Metric ◽  
Geometric Properties ◽  
Unit Vector Field ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Slant Curves

Let (M,g) be a surface and let (U(TM),G) be the unit tangent bundle of M endowed with the Sasaki metric. We know that any curve Γ(s) in U(TM) consist of a curve γ(s) in M and as unit vector field X(s) along γ(s). In this paper we study the geometric properties γ(s) and X(s) satisfying when Γ(s) is a slant geodesic.

Geodesics on the unit tangent bundle

2003 ◽  
Vol 133 (6) ◽  
pp. 1209-1229 ◽  
Author(s):  
J. Berndt ◽  
E. Boeckx ◽  
P. T. Nagy ◽  
L. Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sphere Bundle ◽  
Sasaki Metric ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Unit Tangent Bundle ◽  
Unit Tangent Sphere Bundle

A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.

scholarly journals N-Legendre and N-slant curves in the unit tangent bundle of Minkowski surfaces

2017 ◽  
Vol 11 (01) ◽  
pp. 1850008 ◽  
Author(s):  
Murat Bekar ◽  
Fouzi Hathout ◽  
Yusuf Yayli
Keyword(s):  
Tangent Bundle ◽  
Inner Product ◽  
Normal Vector ◽  
Sasaki Metric ◽  
Reeb Vector ◽  
Nonzero Constant ◽  
Unit Tangent Bundle ◽  
Slant Curves

Let [Formula: see text] be a unit tangent bundle of Minkowski surface [Formula: see text] endowed with the pseudo-Riemannian induced Sasaki metric. In this present paper, we studied the N-Legendre and N-slant curves in which the inner product of its normal vector and Reeb vector is zero and nonzero constant, respectively, in [Formula: see text] and several important characterizations of these curves are given.

scholarly journals Natural Paracontact Magnetic Trajectories on Unit Tangent Bundles

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 72
Author(s):  
Mohamed Tahar Kadaoui Abbassi ◽  
Noura Amri
Keyword(s):  
Gaussian Curvature ◽  
Tangent Bundle ◽  
Two Dimensional ◽  
Constant Gaussian Curvature ◽  
Metric Structures ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Full Classification ◽  
Kaluza Klein

In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics).

scholarly journals Knots connected by wide ribbons

2019 ◽  
Vol 28 (12) ◽  
pp. 1950071
Author(s):  
Susan C. Brooks ◽  
Oguz Durumeric ◽  
Jonathan Simon
Keyword(s):  
Vector Field ◽  
Constant Width ◽  
Unit Vector ◽  
Outer Edge ◽  
Smooth Mapping ◽  
Unit Vector Field ◽  
Finite Set

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus [Formula: see text] in 3-space having constant width [Formula: see text]. Given a regular parametrization [Formula: see text], and a smooth unit vector field [Formula: see text] based along [Formula: see text], for a knot [Formula: see text], we may define a ribbon of width [Formula: see text] associated to [Formula: see text] and [Formula: see text] as the set of all points [Formula: see text], [Formula: see text]. For large [Formula: see text], ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge [Formula: see text] relates to that of the original knot [Formula: see text]. Generically, as [Formula: see text], there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field [Formula: see text]. The particular knot type within the finite set depends on the parametrized curves [Formula: see text], [Formula: see text], and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types [Formula: see text] and [Formula: see text], we can find a smooth ribbon of constant width connecting curves of these two knot types.

scholarly journals A new quaternion valued frame of curves with an application

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 315-330
Author(s):  
Gizem Cansu ◽  
Yusuf Yaylı ◽  
İsmail Gök
Keyword(s):  
Vector Field ◽  
Unit Vector ◽  
Space Curve ◽  
Unit Vector Field ◽  
The Right

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

scholarly journals Geodesic flows on manifolds of negative curvature with smooth horospheric foliations

1991 ◽  
Vol 11 (4) ◽  
pp. 653-686 ◽  
Author(s):  
Renato Feres
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Negative Sectional Curvature

AbstractWe improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

Gauge freedom, anholonomy, and Hopf index of a three-dimensional unit vector field

1993 ◽  
Vol 47 (9) ◽  
pp. 5438-5441 ◽  
Author(s):  
Radha Balakrishnan ◽  
A. R. Bishop ◽  
R. Dandoloff
Keyword(s):  
Vector Field ◽  
Three Dimensional ◽  
Unit Vector ◽  
Gauge Freedom ◽  
Unit Vector Field ◽  
Dimensional Unit

scholarly journals A Ginzburg–Landau Type Energy with Weight and with Convex Potential Near Zero

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 997
Author(s):  
Rejeb Hadiji ◽  
Carmen Perugia
Keyword(s):  
Asymptotic Behavior ◽  
Vector Field ◽  
Lower Bound ◽  
Unit Vector ◽  
Ginzburg Landau ◽  
Positive Weight ◽  
Unit Vector Field

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis–Merle–Rivière.

scholarly journals The Geometry of Tangent Bundles: Canonical Vector Fields

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Tongzhu Li ◽  
Demeter Krupka
Keyword(s):  
Vector Field ◽  
Linear Combination ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Complete Classification ◽  
Constant Coefficients ◽  
Vertical Lift ◽  
Tangent Bundles ◽  
Canonical Vector

A canonical vector field on the tangent bundle is a vector field defined by an invariant coordinate construction. In this paper, a complete classification of canonical vector fields on tangent bundles, depending on vector fields defined on their bases, is obtained. It is shown that every canonical vector field is a linear combination with constant coefficients of three vector fields: the variational vector field (canonical lift), the Liouville vector field, and the vertical lift of a vector field on the base of the tangent bundle.

On the volume of a unit vector field on the three-sphere

1986 ◽  
Vol 61 (1) ◽  
pp. 177-192 ◽  
Author(s):  
Herman Gluck ◽  
Wolfgang Ziller
Keyword(s):  
Vector Field ◽  
Unit Vector ◽  
Three Sphere ◽  
Unit Vector Field
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