scholarly journals Skinning measures in negative curvature and equidistribution of equidistant submanifolds

2013 ◽  
Vol 34 (4) ◽  
pp. 1310-1342 ◽  
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Closed Subset ◽  
Normal Bundle ◽  
Negative Curvature ◽  
Negatively Curved ◽  
Finiteness Result ◽  
Locally Convex ◽  
Outer Unit ◽  
Unit Normal

AbstractLet$C$be a locally convex closed subset of a negatively curved Riemannian manifold$M$. We define the skinning measure${\sigma }_{C} $on the outer unit normal bundle to$C$in$M$by pulling back the Patterson–Sullivan measures at infinity, and give a finiteness result for${\sigma }_{C} $, generalizing the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to$C$equidistribute to the Bowen–Margulis measure${m}_{\mathrm{BM} } $on${T}^{1} M$, assuming only that${m}_{\mathrm{BM} } $is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

scholarly journals Dynamics of the geodesic flow of a foliation

1988 ◽  
Vol 8 (4) ◽  
pp. 637-650 ◽  
Author(s):  
Paweł G. Walczak
Keyword(s):  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Geodesic Flow ◽  
Smooth Measure ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Foliated Riemannian Manifold

AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

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 STABILITY OF ANOSOV HAMILTONIAN STRUCTURES

2010 ◽  
Vol 02 (04) ◽  
pp. 419-451 ◽  
Author(s):  
WILL J. MERRY ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Riemannian Manifold ◽  
Symplectic Form ◽  
Negative Curvature ◽  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Riemannian Metric ◽  
Hamiltonian Structures ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

Let (Mn, g) denote a closed Riemannian manifold (n ≥ 3) which admits a metric of negative curvature (not necessarily equal to g). Let ω1 := ω0 + π*σ denote a twisted symplectic form on TM, where σ ∈ Ω2(M) is a closed 2-form and ω0 is the symplectic structure on TM obtained by pulling back the canonical symplectic form dx ∧ dp on T*M via the Riemannian metric. Let Σk be the hypersurface [Formula: see text]. We prove that if n is odd and the Hamiltonian structure (Σk, ω1) is Anosov with C1 weak bundles, then (Σk, ω1) is stable if and only if it is contact. If n is even and in addition the Hamiltonian structure is 1/2-pinched, then the same conclusion holds. As a corollary, we deduce that if g is negatively curved, strictly 1/4-pinched and σ is not exact then the Hamiltonian structure (Σk, ω1) is never stable for all sufficiently large k.

Flots magnétiques en courbure négative

1999 ◽  
Vol 19 (2) ◽  
pp. 413-436 ◽  
Author(s):  
STÉPHANE GROGNET
Keyword(s):  
Magnetic Field ◽  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Closed Surface ◽  
Magnetic Flow ◽  
Closed Riemannian Manifold ◽  
Dynamical Properties ◽  
Geometric Study

Consider a magnetic field on a closed Riemannian manifold of negative curvature. A geometric study provides dynamical properties of the associated flow stronger than expected for general perturbations of the geodesic flow. Under natural assumptions, a magnetic flow on a closed surface cannot be ${\mathcal C}^1$-conjugate to a geodesic flow.

scholarly journals Entropy estimates for geodesic flows

1982 ◽  
Vol 2 (3-4) ◽  
pp. 513-524 ◽  
Author(s):  
P. Sarnak
Keyword(s):  
Riemannian Manifold ◽  
Topological Entropy ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Compact Riemannian Manifold ◽  
Geodesic Flows ◽  
Unit Tangent Bundle ◽  
Entropy Estimates ◽  
Measure Entropy

AbstractLet M be a compact Riemannian manifold of (variable) negative curvature. Let h be the topological entropy and hμ the measure entropy for the geodesic flow on the unit tangent bundle to M. Estimates for h and hμ in terms of the ‘geometry’ of M are derived. Connections with and applications to other geometric questions are discussed.

Torus actions, maximality, and non-negative curvature

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Escher ◽  
Catherine Searle
Keyword(s):  
Negative Curvature ◽  
Torus Action ◽  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Product Of Spheres ◽  
Rank Conjecture ◽  
Simply Connected ◽  
Symmetry Rank ◽  
Special Case

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Geodesic Flow on Ideal Polyhedra

1997 ◽  
Vol 49 (4) ◽  
pp. 696-707 ◽  
Author(s):  
Charalambos Charitos ◽  
Georgios Tsapogas
Keyword(s):  
Geodesic Flow ◽  
Negative Curvature ◽  
Closed Orbits

AbstractIn this work we study the geodesic flow on n-dimensional ideal polyhedra and establish classical (for manifolds of negative curvature) results concerning the distribution of closed orbits of the flow.

scholarly journals Extension of the unit normal vector field from a hypersurface

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Roland Duduchava ◽  
Eugene Shargorodsky ◽  
George Tephnadze
Keyword(s):  
Vector Field ◽  
Existence And Uniqueness ◽  
Elementary Proof ◽  
Gradient Field ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field ◽  
Outer Unit ◽  
Unit Normal

AbstractIn many applications it is important to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of this paper is to provide an elementary proof of the existence and uniqueness of such an extension.

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