Topological stability and Gromov hyperbolicity

1999 ◽  
Vol 19 (1) ◽  
pp. 143-154 ◽  
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Hyperbolic Space ◽  
Geodesic Flow ◽  
Positive Curvature ◽  
Universal Covering ◽  
Conjugate Points ◽  
Gromov Hyperbolicity ◽  
Topological Stability ◽  
Shadowing Property ◽  
Gromov Hyperbolic

We show that if the geodesic flow of a compact analytic Riemannian manifold $M$ of non-positive curvature is either $C^{k}$-topologically stable or satisfies the $\epsilon$-$C^{k}$-shadowing property for some $k > 0$ then the universal covering of $M$ is a Gromov hyperbolic space. The same holds for compact surfaces without conjugate points.

Weak stability of the geodesic flow and Preissmann's theorem

2000 ◽  
Vol 20 (4) ◽  
pp. 1231-1251
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Geodesic Flow ◽  
Abelian Subgroup ◽  
Conjugate Points ◽  
Weak Stability ◽  
Shadowing Property

Let $(M,g)$ be a compact, differentiable Riemannian manifold without conjugate points and bounded asymptote. We show that, if the geodesic flow of $(M,g)$ is either topologically stable, or satisfies the $\epsilon$-shadowing property for some appropriate $\epsilon > 0$, then every abelian subgroup of the fundamental group of $M$ is infinite cyclic. The proof is based on the existence of homoclinic geodesics in perturbations of $(M,g)$, whenever there is a subgroup of the fundamental group of $M$ isomorphic to $\mathbb{Z}\times \mathbb{Z}$.

On a conjecture about expansive geodesic flows

1996 ◽  
Vol 16 (3) ◽  
pp. 545-553 ◽  
Author(s):  
Rafael Oswaldo Ruggierot
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Conjugate Points ◽  
Geodesic Flows

AbstractWe show that near the geodesic flow of a compact Riemannian manifold with no conjugate points which is expansive, every expansive geodesic flow has no conjugate points. We also prove that in the above hypotheses the geodesic flow istopologically stable.

The accessibility property of expansive geodesic flows without conjugate points

2008 ◽  
Vol 28 (1) ◽  
pp. 229-244
Author(s):  
RAFAEL OSWALDO RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Conjugate Points ◽  
Stable Set ◽  
Geodesic Flows ◽  
Continuous Path ◽  
Two Dimensional ◽  
Unstable Set ◽  
Unit Tangent Bundle ◽  
Continuous Curves

AbstractLet (M,g) be a compact, smooth Riemannian manifold without conjugate points whose geodesic flow is expansive. We show that the geodesic flow of (M,g) has the accessibility property, namely, given two pointsθ1,θ2in the unit tangent bundle, there exists a continuous path joiningθ1,θ2formed by the union of a finite number of continuous curves, each of which is contained either in a strong stable set or in a strong unstable set of the dynamics. Since expansive geodesic flows of compact surfaces have no conjugate points, the accessibility property holds for every two-dimensional expansive geodesic flow.

scholarly journals Energy-minimizing maps from manifolds with nonnegative Ricci curvature

2019 ◽  
pp. 1950083 ◽  
Author(s):  
James Dibble
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Universal Covering ◽  
Complete Riemannian Manifold ◽  
Conjugate Points ◽  
Nonnegative Ricci Curvature ◽  
Universal Covering Space ◽  
Asymptotic Geometry ◽  
Bounded Below ◽  
Minimizing Maps

The energy of any [Formula: see text] representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only if the original map is totally geodesic. This conclusion also holds under the weaker assumption that the domain is finitely covered by a diffeomorphic product, and its universal covering space splits isometrically as a product with a flat factor, in a commutative diagram that follows from the Cheeger–Gromoll splitting theorem.

Expansive geodesic flows in manifolds with no conjugate points

1997 ◽  
Vol 17 (1) ◽  
pp. 211-225 ◽  
Author(s):  
RAFAEL O. RUGGIERO
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Product Structure ◽  
Conjugate Points ◽  
Geodesic Flows ◽  
Closed Geodesics ◽  
Local Product ◽  
Unit Tangent Bundle

Let $M$ be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. We show that there exists a local product structure in the unit tangent bundle of the manifold which is invariant under the geodesic flow. In particular, we have that the set of closed geodesics is dense and that the flow is topologically transitive.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

Expansive geodesic flows on surfaces

1993 ◽  
Vol 13 (1) ◽  
pp. 153-165 ◽  
Author(s):  
Miguel Paternain
Keyword(s):  
Geodesic Flow ◽  
Compact Surface ◽  
Conjugate Points ◽  
Riemannian Surface ◽  
Geodesic Flows ◽  
Flows On Surfaces

AbstractWe prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.

scholarly journals Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

2018 ◽  
Vol 2020 (5) ◽  
pp. 1346-1365 ◽  
Author(s):  
Jason DeVito ◽  
Ezra Nance
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Lie Group ◽  
Positive Curvature ◽  
Positive Sectional Curvature ◽  
Compact Symmetric Space ◽  
Rank 2 ◽  
Set Of Points ◽  
Positively Curved

Abstract A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

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