scholarly journals Marked boundary rigidity for surfaces

2016 ◽  
Vol 38 (4) ◽  
pp. 1459-1478 ◽  
Author(s):  
COLIN GUILLARMOU ◽  
MARCO MAZZUCCHELLI
Keyword(s):  
Riemannian Metrics ◽  
Compact Surface ◽  
Conjugate Points ◽  
Geodesic Flows ◽  
Convex Boundary ◽  
Strictly Convex ◽  
Negatively Curved ◽  
Boundary Rigidity ◽  
Boundary Distance

We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

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.

Harmonic functions on ℝ-covered foliations

2009 ◽  
Vol 29 (4) ◽  
pp. 1141-1161
Author(s):  
S. FENLEY ◽  
R. FERES ◽  
K. PARWANI
Keyword(s):  
Harmonic Functions ◽  
Riemannian Metrics ◽  
Related Result ◽  
Continuous Functions ◽  
Property A ◽  
Liouville Property ◽  
Foliated Manifold ◽  
Curved Manifolds ◽  
Codimension One ◽  
Negatively Curved

AbstractLet (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

scholarly journals Marked length rigidity for Fuchsian buildings

2018 ◽  
Vol 39 (12) ◽  
pp. 3262-3291
Author(s):  
DAVID CONSTANTINE ◽  
JEAN-FRANÇOIS LAFONT
Keyword(s):  
Automorphism Group ◽  
Riemannian Metrics ◽  
Length Spectrum ◽  
Negatively Curved ◽  
Spectrum Function ◽  
Marked Length Spectrum

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT($-1$) metrics on $X$, which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.

scholarly journals Approximate solutions of cohomological equations associated with some Anosov flows

1990 ◽  
Vol 10 (2) ◽  
pp. 367-379 ◽  
Author(s):  
Svetlana Katok
Keyword(s):  
Approximate Solutions ◽  
Geodesic Flows ◽  
Curved Surfaces ◽  
Anosov Flow ◽  
3 Dimensional ◽  
Negatively Curved ◽  
Anosov Flows ◽  
Higher Differentiability ◽  
Special Case ◽  
Cohomological Equations

AbstractThe Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.

scholarly journals On Strictly Convex Subsets in Negatively Curved Manifolds

2010 ◽  
Vol 22 (3) ◽  
pp. 621-632 ◽  
Author(s):  
Jouni Parkkonen ◽  
Frédéric Paulin
Keyword(s):  
Strictly Convex ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Convex Subsets

scholarly journals Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics

2016 ◽  
Vol 20 (1) ◽  
pp. 469-490 ◽  
Author(s):  
Dmitri Burago ◽  
Sergei Ivanov
Keyword(s):  
Finsler Metrics ◽  
Geodesic Flows ◽  
Boundary Distance

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.

Regularity at infinity of compact negatively curved manifolds

1994 ◽  
Vol 14 (3) ◽  
pp. 493-514
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Tangent Bundle ◽  
Compact Manifold ◽  
Negative Curvature ◽  
Periodic Points ◽  
Geodesic Flows ◽  
Stable Foliation ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Unit Tangent Bundle ◽  
Measure Class

AbstractIt is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.

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