scholarly journals Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane

2016 ◽  
Vol 71 (3-4) ◽  
pp. 1473-1489 ◽  
Author(s):  
Masatomo Takahashi
Keyword(s):  
Tangent Bundle ◽  
Euclidean Plane ◽  
Unit Tangent Bundle ◽  
Legendre Curves

scholarly journals Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame

2021 ◽  
Vol 73 (5) ◽  
pp. 589-601
Author(s):  
M. Bekar ◽  
F. Hathout ◽  
Y. Yayli
Keyword(s):  
Tangent Bundle ◽  
Vector Fields ◽  
Ruled Surfaces ◽  
Unit Tangent Bundle ◽  
Legendre Curves ◽  
Rotation Minimizing Frame

UDC 514.7 In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified.

scholarly journals Framed curves in the Euclidean space

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
Shun’ichi Honda ◽  
Masatomo Takahashi
Keyword(s):  
Euclidean Space ◽  
Tangent Bundle ◽  
Moving Frame ◽  
Regular Curve ◽  
Smooth Functions ◽  
Independent Condition ◽  
Legendre Curve ◽  
Space Curves ◽  
Unit Tangent Bundle ◽  
Legendre Curves

AbstractA framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

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 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.

A new description of the Bowen—Margulis measure

1989 ◽  
Vol 9 (3) ◽  
pp. 455-464 ◽  
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Unstable Manifold ◽  
Tangent Bundle ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Universal Covering ◽  
Unstable Foliation ◽  
Unit Tangent Bundle

AbstractThe Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.

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 Density of half-horocycles on geometrically infinite hyperbolic surfaces

2012 ◽  
Vol 33 (4) ◽  
pp. 1162-1177
Author(s):  
BARBARA SCHAPIRA
Keyword(s):  
Tangent Bundle ◽  
Hyperbolic Surface ◽  
Hyperbolic Surfaces ◽  
Weak Assumption ◽  
Unit Tangent Bundle

AbstractOn the unit tangent bundle of a hyperbolic surface, we study the density of positive orbits $(h^s v)_{s\ge 0}$ under the horocyclic flow. More precisely, given a full orbit $(h^sv)_{s\in {\mathbb R}}$, we prove that under a weak assumption on the vector $v$, both half-orbits $(h^sv)_{s\ge 0}$ and $(h^s v)_{s\le 0}$ are simultaneously dense or not in the non-wandering set $\mathcal {E}$of the horocyclic flow. We give also a counterexample to this result when this assumption is not satisfied.

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