When is an Anosov flow geodesic?

1992 ◽  
Vol 12 (2) ◽  
pp. 227-232
Author(s):  
Leon W. Green
Keyword(s):  
Flow Field ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Anosov Flow ◽  
Unit Tangent Bundle

AbstractLet X, H+, H− be vector fields tangent, respectively, to an Anosov flow and its expanding and contracting foliations in a compact three-dimensional manifold, with γ, α+, α− one forms dual to them. If α+([H+, H−]) = α−([H+, H−]) and γ([H+, H−]) = α−([X, H−]) − α+([X, H+]), then the manifold has the structure of the unit tangent bundle of a Riemannian orbifold with geodesic flow field X.

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 Coding of geodesics and Lorenz-like templates for some geodesic flows

2016 ◽  
Vol 38 (3) ◽  
pp. 940-960
Author(s):  
PIERRE DEHORNOY ◽  
TALI PINSKY
Keyword(s):  
Periodic Orbits ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Hyperbolic Metric ◽  
Geodesic Flows ◽  
Unit Tangent Bundle

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

Lefschetz formulae for Anosov flows on 3-manifolds

1993 ◽  
Vol 13 (2) ◽  
pp. 335-347 ◽  
Author(s):  
Héctor Sánchez-Morgado
Keyword(s):  
Fundamental Group ◽  
Unitary Representation ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Constant Negative Curvature ◽  
Closed Orbits ◽  
Unit Tangent Bundle ◽  
Anosov Flows

AbstractFried has related closed orbits of the geodesic flow of a surface S of constant negative curvature to the R-torsion for a unitary representation of the fundamental group of the unit tangent bundle T1S. In this paper we extend those results to transitive Anosov flows and 2-dimensional attractors on 3-manifolds.

scholarly journals Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature

1990 ◽  
Vol 10 (4) ◽  
pp. 657-670 ◽  
Author(s):  
Renato Feres ◽  
Anatole Katok
Keyword(s):  
Geodesic Flow ◽  
Negative Curvature ◽  
Flow Direction ◽  
Torsion Tensor ◽  
Affine Connection ◽  
Invariant Probability Measure ◽  
Three Dimensional ◽  
Ergodic Measure ◽  
Dimensional Manifold ◽  
Anosov Flows

AbstractWe consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C∞ affine connection ∇ on V such that ∇R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V) and for any invariant ergodic measure μ, there exists χμ > 0 such that the Lyapunov exponents are −2χμ, −χμ, 0, χμ, and 2χμ, μ-almost everywhere.As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C∞. Then, this flow is C∞ conjugate to the geodesic flow on a manifold of constant negative curvature.

scholarly journals Complete lift of vector fields and sprays to T∞M

2015 ◽  
Vol 12 (10) ◽  
pp. 1550113 ◽  
Author(s):  
Ali Suri ◽  
Somaye Rastegarzadeh
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Dimensional Manifold ◽  
Closed Curves ◽  
Infinite Dimensional ◽  
Integral Curves ◽  
Complete Lift ◽  
Infinite Dimensional Manifold

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

scholarly journals Visualization of Post-Processed CFD Data in a Virtual Environment

1999 ◽  
Author(s):  
Vishant J. Shahnawaz ◽  
Judy M. Vance ◽  
Sasikumar V. Kutti
Keyword(s):  
Virtual Reality ◽  
Fluid Flow ◽  
Flow Field ◽  
Vector Fields ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Scalar Fields ◽  
Grid Data ◽  
Head Mounted Display ◽  
Graphical Environment

Abstract This paper discusses the development of a virtual reality (VR) interface for the visualization of Computational Fluid Dynamics (CFD) data. The application, VR-CFD, provides an immersive and interactive graphical environment in which users can examine the analysis results from a CFD analysis of a flow field in three-dimensional space. It has been tested and implemented with virtual reality devices such as the C2, head mounted display (HMD) and desktop VR. The application is designed to read PLOT3D structured grid data and to display the flow field parameters using features such as streamlines, cutting planes, iso-surfaces, rakes, vector fields and scalar fields. Visualization Toolkit (VTK), a data visualization library, is used along with OpenGL arid the C2 VR interface libraries, to develop the application. Analysts and designers have used VR-CFD to visualize and understand complex three-dimensional fluid flow phenomena. The combination of three-dimensional interaction capability and the C2 virtual reality environment has been shown to facilitate collaborative discussions between analysts and engineers concerning the appropriateness of the CFD model and the characteristics of the fluid flow.

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.

scholarly journals Genus-one Birkhoff sections for geodesic flows

2014 ◽  
Vol 35 (6) ◽  
pp. 1795-1813 ◽  
Author(s):  
PIERRE DEHORNOY
Keyword(s):  
Tangent Bundle ◽  
Geodesic Flow ◽  
Singular Points ◽  
Geodesic Flows ◽  
Return Maps ◽  
Unit Tangent Bundle ◽  
First Return Maps ◽  
Genus One

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with three or four singular points admits explicit genus-one Birkhoff sections, and we determine the associated first return maps.

scholarly journals Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

2021 ◽  
pp. 1-10
Author(s):  
ALINE CERQUEIRA ◽  
CARLOS G. MOREIRA ◽  
SERGIO ROMAÑA
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Riemannian Metric ◽  
Hausdorff Dimensions ◽  
Hyperbolic Set ◽  
Negatively Curved ◽  
Smooth Perturbation ◽  
Unit Tangent Bundle ◽  
Markov Spectrum

Abstract Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

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