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Les géométries de Hilbert sont à géométrie locale bornée

Bruno Colbois, Constantin Vernicos (2007)

Annales de l’institut Fourier

On montre que la géométrie de Hilbert d’un domaine convexe de ℝ n est à géométrie locale bornée c-à-d que pour un rayon fixé, toutes les boules sont bilipschitz à un domaine de ℝ n euclidien. On en déduit que si la géométrie de Hilbert est hyperbolique au sens de Gromov, alors le bas de son spectre est strictement positif. On donne un contre-exemple en dimension trois qui montre que la réciproque n’est pas vraie pour les géométries de Hilbert non planes.

Local coordinates for SL ( n , C ) -character varieties of finite-volume hyperbolic 3-manifolds

Pere Menal-Ferrer, Joan Porti (2012)

Annales mathématiques Blaise Pascal

Given a finite-volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL ( 2 , C ) with the n -dimensional irreducible representation of SL ( 2 , C ) in SL ( n , C ) . In this paper we give local coordinates of the SL ( n , C ) -character variety around the character of this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.

Microlocal Approach to Tensor Tomography and Boundary and Lens Rigidity

Stefanov, Plamen (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 53C24, 53C65, 53C21.This is a survey of the recent results by the author and Gunther Uhlmann on the boundary rigidity problem and on the associated tensor tomography problem.Author partly supported by NSF Grant DMS-0400869.

On g -natural conformal vector fields on unit tangent bundles

Mohamed Tahar Kadaoui Abbassi, Noura Amri (2021)

Czechoslovak Mathematical Journal

We study conformal and Killing vector fields on the unit tangent bundle, over a Riemannian manifold, equipped with an arbitrary pseudo-Riemannian g -natural metric. We characterize the conformal and Killing conditions for classical lifts of vector fields and we give a full classification of conformal fiber-preserving vector fields on the unit tangent bundle endowed with an arbitrary pseudo-Riemannian Kaluza-Klein type metric.

On local isometric immersions into complex and quaternionic projective spaces

Hans Jakob Rivertz (2011)

Archivum Mathematicum

We will prove that if an open subset of ℂ P n is isometrically immersed into ℂ P m , with m < ( 4 / 3 ) n - 2 / 3 , then the image is totally geodesic. We will also prove that if an open subset of ℍ P n isometrically immersed into ℍ P m , with m < ( 4 / 3 ) n - 5 / 6 , then the image is totally geodesic.

On rank one symmetric space

Inkang Kim (2004/2005)

Séminaire de théorie spectrale et géométrie

In this paper we survey some recent results on rank one symmetric space.

On the completeness of total spaces of horizontally conformal submersions

Mohamed Tahar Kadaoui Abbassi, Ibrahim Lakrini (2021)

Communications in Mathematics

In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class...

On the geometry of frame bundles

Kamil Niedziałomski (2012)

Archivum Mathematicum

Let ( M , g ) be a Riemannian manifold, L ( M ) its frame bundle. We construct new examples of Riemannian metrics, which are obtained from Riemannian metrics on the tangent bundle T M . We compute the Levi–Civita connection and curvatures of these metrics.

On the Hausdorff Dimension of CAT(κ) Surfaces

David Constantine, Jean-François Lafont (2016)

Analysis and Geometry in Metric Spaces

We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.

Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

François Fillastre (2007)

Annales de l’institut Fourier

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus greater than one. We prove that these metrics are actually realised by exactly one convex...

Riemannian foliations with parallel or harmonic basic forms

Fida El Chami, Georges Habib, Roger Nakad (2015)

Archivum Mathematicum

In this paper, we consider a Riemannian foliation that admits a nontrivial parallel or harmonic basic form. We estimate the norm of the O’Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.

Rigidity and L 2 cohomology of hyperbolic manifolds

Gilles Carron (2010)

Annales de l’institut Fourier

When X = Γ ∖ ℍ n is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L 2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.

Rigidity at infinity for even-dimensional asymptotically complex hyperbolic spaces

Hassan Boualem, Marc Herzlich (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Any Kähler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.

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