Metric Spaces, Convexity and Nonpositive Curvature

Papadopoulos, Athanase

doi:10.4171/132

Cited by 152 publications

(143 citation statements)

References 4 publications

Supporting

Mentioning

139

Contrasting

Unclassified

Order By: Relevance

“…We begin with some standard notation and basic facts about geodesic spaces, which we use in the sequel; one may find a much more through description of the concepts presented below, for example, in [2] and [13]. Let us assume that a metric space (X, ρ) is geodesic, which means that each couple of points can be joined by a geodesic, i.e., an isometric embedding γ :…”

Section: Preliminariesmentioning

confidence: 99%

On the geometry and fixed point free mappings in hyperbolic spaces

Piątek

2018

J. Fixed Point Theory Appl.

View full text Add to dashboard Cite

Abstract. In the paper, we focus on two problems in geodesic hyperbolic spaces. First, we will consider the behavior of Picard iterative sequences of fixed point free nonexpansive mappings in spaces of constant curvature. Second, motivated by Hotchkiss (Proc Am Math Soc 125:1903-1912, 1997 we will show under which additional assumptions the Gromov boundary of a hyperbolic space equipped with the topology defined by the visual metric coincides with the geodesic boundary equipped with the cone topology.Mathematics Subject Classification. Primary 47H09, 53C23; Secondary 47H10, 53C22.

show abstract

Section: Preliminariesmentioning

confidence: 99%

On the geometry and fixed point free mappings in hyperbolic spaces

Piątek

2018

J. Fixed Point Theory Appl.

View full text Add to dashboard Cite

show abstract

“…We refer here on the books [1], [3], [4] and [8] only. These books contain a vast old and up-to-date bibliography of works on different aspects of the geometry of distance and Finsler spaces.…”

Section: Preliminariesmentioning

confidence: 99%

Differentiable Distance Spaces

Kertész

Tamássy

2015

Acta Math. Hungar.

View full text Add to dashboard Cite

The distance function ̺(p, q) (or d(p, q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over R n , whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations.

show abstract

“…We refer the reader to [44] for a systematic presentation of several properties of Busemann functions.…”

Section: Strictly Convex Manifolds and Topological Entropymentioning

confidence: 99%

Convex real projective structures and Hilbert metrics

Ким

Papadopoulos

2014

IRMA Lectures in Mathematics and Theoretical Physics

Self Cite

View full text Add to dashboard Cite

Abstract. We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel-Nielsen parameters of Teichmüller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in SL(3, R). In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichmüller space of the surface.

show abstract

Metric Spaces, Convexity and Nonpositive Curvature

Cited by 152 publications

References 4 publications

On the geometry and fixed point free mappings in hyperbolic spaces

On the geometry and fixed point free mappings in hyperbolic spaces

Differentiable Distance Spaces

Convex real projective structures and Hilbert metrics

Contact Info

Product

Resources

About