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2 edition of Hyperbolic behaviour of geodesic flows on manifolds with no focal points found in the catalog.

Hyperbolic behaviour of geodesic flows on manifolds with no focal points

Keith Burns

# Hyperbolic behaviour of geodesic flows on manifolds with no focal points

Written in English

Edition Notes

Thesis (Ph.D.) - University of Warwick, 1983.

 ID Numbers Statement Keith Burns. Open Library OL14830203M

These arise naturally as the boundary of hyperbolic 3-manifolds and will play an im-portant role in the extension of the Hodgson-Kerckhoffdeformation theory to innite volume and geometrically nite hyperbolic cone-manifolds. One-parameter families of metrics We start with a family of metrics, gt: V V! R, on a nite dimensional vector space V. The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces Benjamin Linowitz, Je rey S. Meyer and Paul Pollack Abstract. In this paper we examine the relationship between the length spec-trum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M.

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### Hyperbolic behaviour of geodesic flows on manifolds with no focal points by Keith Burns Download PDF EPUB FB2

Hyperbolic behaviour of geodesic flows on manifolds with no focal points Keith Burns Ergodic Theory and Dynamical Systems 3. Crossref. On the ergodicity of geodesic flows W. Ballmann and M. Brin Ergodic Theory and Dynamical Systems 2.

Crossref. Geodesic flows with hyperbolic behaviour of the trajectories and objects connected with themCited by: In this article, we investigate the transitivity of geodesic flows on connected compact rank 1 manifolds without focal points.

Let (M, g) be a C ∞ connected compact n-dimensional Riemannian manifold, where g is a Riemannian metric. For any p ∈ M and v ∈ T p M, let γ v be the unique geodesic satisfying the initial conditions γ v (0) = p Cited by: 2. Hyperbolic, at, and elliptic manifolds 49 Complete hyperbolic manifolds 50 Discrete groups 51 Hyperbolic manifolds with boundary 52 Non-complete hyperbolic manifolds 53 2.

Polyhedra 54 Polyhedra and tessellations 54 Voronoi tessellations 55 Fundamental and Dirichlet domains 56 Flat manifolds 57 File Size: 3MB.

ON THE STABILITY CONJECTURE FOR GEODESIC FLOWS OF MANIFOLDS WITHOUT CONJUGATE POINTS L. RIFFORD AND R. RUGGIERO Abstract. We study the C2-structural stability conjecture from Man~ e’s view-point for geodesics ows of compact manifolds without conjugate points.

The structural stability conjecture is an open problem in the category of geodesicFile Size: KB. In this article, we consider the entropy-expansiveness of geodesic flows on closed Riemannian manifolds without conjugate points. We prove that, if the manifold has no focal points, or if the.

Totally geodesic maps into manifolds with no focal points BY JAMES DIBBLE Dissertation Director: Xiaochun Rong The space of totally geodesic maps in each homotopy class [F] from a compact Riemannian man-ifold M with non-negative Ricci curvature into a complete Riemannian manifold N with no focal points is path-connected.

the constant of motion $-H_1^{-1/2}H_2$ describes the Hyperbolic behaviour of geodesic flows on manifolds with no focal points book curvature of the geodesics for the hyperbolic plane.

You could find interesting this set of notes. It shows two examples of completely integrable geodesic flows: the one on the hyperbolic plane and the other on the revolution surfaces; it appears to me to be enough detailed.

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any formally, the exponential map at point p, is defined on T p M, the entire tangent space at p.

Equivalently, consider a maximal geodesic: →. conformal unstable manifolds is sketched in the Appendix in Chapter 7. A large part of Chapter 3, 5, 6, 7 overlap the one in the two preprints [56], [57]. Geodesic ows on manifolds of non-positive curvature The geodesic ow on compact manifolds of strictly negative curvature is a primary example of uniformly hyperbolic systems.

Its ergodic prop-Author: Weisheng Wu. Structure of manifolds of nonpositive sectional curvature. harmonic manifolds without focal points or with Gromov hyperbolic fundamental groups. of geodesic flows on manifolds with no Author: Werner Ballmann.

Hyperbolic geometry is as real to mathematicians as euclidean. It could even be that the universe is a hyperbolic space, as analysis of the data on the cosmic microwave background (CMB) by Aurich, Lustig, Steiner, Then [ALST] has suggested.

Hyperbolic surfaces and their geodesics Spaces modelled on the hyperbolic disc are called hyperbolic File Size: KB. Description The space of totally geodesic maps in each homotopy class [F] from a compact Riemannian manifold M with non-negative Ricci curvature into a complete Riemannian manifold N with no focal points is path-connected.

If [F] contains Hyperbolic behaviour of geodesic flows on manifolds with no focal points book totally geodesic map, then each map in [F] is energy-minimizing if and only if it is totally geodesic. When N is compact, each map from a product W x M.

POWER SPECTRUM OF THE GEODESIC FLOW ON HYPERBOLIC MANIFOLDS the hyperbolic space HnC1 appearing as the leading coefﬁcient of a weak asymptotic expansion at Sn of the lift of f to HnC1. Then u is described by wvia an explicit formula,(); this formula features the Poisson kernel P and the map B VSHnC1.

K. BURNS, "Hyperbolic behaviour of geodesic flows on manifolds with no focal points", Erg. Dyn. Syst. 3 (), 1– MathSciNet CrossRef zbMATH Google Scholar [BS]Cited by: 1. This was answered in generality by D.V. Anosov () in his paper Geodesic flows on closed Riemannian manifolds of negative curvature.A free copy is available online, but it is somewhat long and written entirely in Russian.

With regard to tracking this down: After learning of earlier results due to Hopf, I searched Google books for "hopf argument" "negative curvature". geodesic hyperbolic 3–manifolds in ﬁnite covers. However, until recently, no single example of a hyperbolic 3–manifold that did bound geometrically was known; the ﬁrst example was given [16].

(3) The smallest known hyperbolic 3–manifold with η(M) ∈ Z is the manifold. Burns, Hyperbolic behaviour of geodesic flows on manifolds with no focal points, Ergodic Theory Dynam. Systems, 3 (), doi: /S Google Scholar [14] K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Inst.

Hautes Etudes Sci. Publ. Math., 65 (), Google ScholarCited by: 2. The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds Huagui Duan1, Yiming Long2,y Wei Wang3 z 1 School of Mathematical Sciences and LPMC, Nankai University, Tianjin 2 Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 3 School of Mathematical Sciences and LMAM, Peking University, Beijing File Size: KB.

We give a short and direct proof of exponential mixing of geodesic flows on compact hyperbolic three-manifolds with respect to the Liouville measure. This complements earlier results of Collet-Epstein-Gallovotti, Moore, and Ratner for hyperbolic surfaces.

Furthermore, since the analysis is even easier in three dimensions than in two dimensions (because of the absence of discrete series Cited by: A geodesic in a Riemannian manifold is a locally distance-minimising curve, and is said to be simple if it has no self-intersections and nonsimple otherwise.

Thus the simple closed geodesics in a Riemannian manifold are precisely its geodesic knots. This thesis is a study of geodesic knots in hyperbolic 3-manifolds. By a hy. Buy Geodesic flows on closed Riemann manifolds with negative curvature, (Proceedings of the Steklov Institute of Mathematics, no.

90, ) on FREE SHIPPING on qualified ordersAuthor: D. V Anosov. geodesic planes in hyperbolic 3-manifolds M of in nite volume. In the case of an acylindrical 3-manifold whose convex core has totally geodesic boundary, we show that the closure of any geodesic plane is a properly immersed submanifold of M.

On the other hand, we show that rigidity fails for quasifuchsian manifolds. Contents. Hi All, I am trying to figure out the details on giving a surface S a hyperbolic metric with geodesic boundary, i.e., a metric of constant sectional curvature -1 so that the (manifold) boundary components, i.e., a collection of disjoint simple-closed curves are geodesics under this metric.

The simplicial volume of hyperbolic manifolds with geodesic boundary Item Preview The simplicial volume of hyperbolic manifolds with geodesic boundary by Roberto Frigerio thus providing a somewhat new proof of the proportionality principle for non-compact finite-volume hyperbolic n-manifolds without boundary.

Addeddate §3. Hyperbolic properties of geodesic flows 18 §4. The axiom of visibility and the axiom of asymptoticity 20 §5. Limiting spheres 28 §6.

Topological properties of geodesic flows 34 §7. Ergodic properties of geodesic flows 36 §8. Geodesic flows on manifolds of Anosov type 40 Part II. Frame flows and horocycle flows 44 § 9. Definition of a. Geodesic flows and Curvature. Ask Question Doesn't this mean that if two points (or rather two nbds around those points) are connected by a geodesic flow then the metric at those points should be the understood to be the "equivalent" of Hopf-Rinow in Lorentzian geometry is the theorem that states that in any globally hyperbolic.

Click on the article title to read : Donal Hurley. In this paper we focus on complete hyperbolic n-dimensional (n 2 3) manifolds of finite volume with non-empty totally geodesic boundary. Such a manifold N has double DN along its boundary 8N, so that DN is a complete hyperbolic n-manifold with finite volume and that dN is totally geodesic in DN.

Geodesic systems of tunnels in hyperbolic 3–manifolds STEPHAN D. BURTON the proof involves constructing a sequence of hyperbolic manifolds with the arc the shortest distance in an –ball about the nearest points on the geodesic will pass through the geodesic, so it will not be an isotopy.) In any case, the geodesic in theFile Size: KB.

Geometric properties of hyperbolic geodesics there exist unique s > 0 and θ in [0,2π) with za(s,θ) = z. The hyperbolic polar coordinates of the point z relative to the center or pole at a are the ordered pair (s,θ), where za(s,θ) = z.

The ﬁrst coordinate, s = dD(a,z), is the hyperbolic. arithmetic hyperbolic n-manifold either contains no codimension k geodesic sub-manifolds, or it contains in nitely many and they are everywhere dense.

This was perhaps rst made precise in dimension 3 by Maclachlan{Reid and Reid [19, 33], who exhibited the rst hyperbolic 3-manifolds with no. Let (M, g) be a compact Riemannian manifold of hyperbolic type without conjugate points and X be its universal Riemannian show that the growth function of the volume of geodesic spheres of X is of purely exponential type.

This result yields a sufficient condition for the non-existence of a Riemannian metric with strictly negative curvature on compact by: 1. A closed geodesic in a (closed) hyperbolic n-manifold is simple if it has no self-intersections, and non-simple otherwise.

In dimension 2 every closed hyperbolic manifold has a non-simple closed geodesic. However in dimension 3 the situation is much more complex. Many closed hyperbolic 3-manifolds contain immersions of totally geodesic surfaces.

manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds.

For hyperbolic 3-manifolds which contain an immersed totally geodesic surfaces there are. I reflected your comments in my edit. I dont think though that $\alpha$ can be taken to be the volume form on $\partial M.$ Cut a closed hyperbolic surface by a non-seperating geodesic.

The surface area will remain a multiple of $\pi,$ while the length of the geodesic may be arbitrary. $\endgroup$ –. Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps.

BABILLOT Martine (1) & PEIGNE Marc (2) To Martine’s memory The redaction of this paper has been overshadowed by Martine’s death in July All the main ideas were worked out together, I have done my best to ﬁnish this paper.

by: Gabriel Pedro Paternain is a Uruguayan is Professor of Mathematics in DPMMS at the University of Cambridge, and a fellow of Trinity obtained his Licenciatura from Universidad de la Republica in Uruguay inand his PhD from the State University of New York at Stony Brook in He has lectured several undergraduate and graduate courses and has gained Doctoral advisor: Detlef Gromoll.

This thesis investigates the topology and geometry of hyperbolic 3-manifolds containing totally geodesic surfaces. In chapter 2 we find algebraic invariants associated to several families of hyperbolic 3-manifolds with totally geodesic boundary. Geodesic flows on closed Riemannian manifolds with negative curvature.

() by D V Anosov rigidity and Godbillon-Vey classes for Anosov flows by S. Hurder, A. Katok We study compact group extensions of hyperbolic dif-feomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable.

stand complete, non-singular hyperbolic 3-manifolds. Hodgson and Kerckho in-troduced analytic techniques to the study of cone-manifolds that they have used to prove deep results about nite volume hyperbolic 3-manifolds.

In this paper we use Hodgson and Kerckho ’s techniques to study in nite volume hyperbolic 3-manifolds. Teichmiiller space r(S) of conformal (or hyperbolic) structures on S. It has been conjec- tured that the locus of these points is related in an approximate way to a geodesic in F(S), and this is known to be true for a class of examples arising from hyperbolic structures on surface bundles over a circle (see [S]).Citation McCullen, Curtis T., Amir Mohammadi, and Hee Oh.

Geodesic planes in hyperbolic 3-manifolds. Working paper, Harvard University.Hyperbolic, at, and elliptic manifolds 43 Discrete groups of isometries 44 Polyhedra and tessellations 46 Fundamental and Dirichlet domains 46 Flat manifolds 47 Elliptic manifolds 48 Selberg lemma 49 Triangular groups 50 Ideal polyhedra 51 Platonic solids 54 2.

Generalities on hyperbolic.