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Author : Athanase Papadopoulos
Publisher : European Mathematical Society
Page : 306 pages
File Size : 48,6 Mb
Release : 2005
Category : Computers
ISBN : 3037190108
Author : Martin R. Bridson,André Häfliger
Publisher : Springer Science & Business Media
Page : 665 pages
File Size : 42,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662124949
A description of the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by isometries. The theory of these objects is developed in a manner accessible to anyone familiar with the rudiments of topology and group theory: non-trivial theorems are proved by concatenating elementary geometric arguments, and many examples are given. Part I provides an introduction to the geometry of geodesic spaces, while Part II develops the basic theory of spaces with upper curvature bounds. More specialized topics, such as complexes of groups, are covered in Part III.
Author : Jürgen Jost
Publisher : Birkhauser
Page : 128 pages
File Size : 46,6 Mb
Release : 1997
Category : Geometry, Differential
ISBN : UCSD:31822023840598
This book discusses various geometric and analytic aspects of nonpositive curvature, starting with a discussion of Riemannian examples and rigidity theorems. It then treats generalized notions of nonpositive curvature in metric geometry in the sense of Alexandrov and Busemann, as well as the theory of harmonic maps with values in such spaces. It is intended for researchers and graduate students in Riemannian and metric geometry as well as calculus of variations.
Author : Werner Ballmann
Publisher : Birkhäuser
Page : 114 pages
File Size : 53,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034892407
Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.
Author : Eugene Zaustinskiy
Publisher : American Mathematical Soc.
Page : 91 pages
File Size : 54,7 Mb
Release : 1959
Category : Distance geometry
ISBN : 9780821812341
Author : Werner Ballmann,Misha Gromov,Viktor Schroeder
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 55,8 Mb
Release : 2013-12-11
Category : Mathematics
ISBN : 9781468491593
This volume presents a complete and self-contained description of new results in the theory of manifolds of nonpositive curvature. It is based on lectures delivered by M. Gromov at the Collège de France in Paris. Therefore this book may also serve as an introduction to the subject of nonpositively curved manifolds. The latest progress in this area is reflected in the article of W. Ballmann describing the structure of manifolds of higher rank.
Author : Stephanie Alexander,Vitali Kapovitch,Anton Petrunin
Publisher : Springer
Page : 88 pages
File Size : 52,7 Mb
Release : 2019-05-08
Category : Mathematics
ISBN : 9783030053123
Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.
Author : Luigi Ambrosio,Nicola Gigli,Giuseppe Savare
Publisher : Springer Science & Business Media
Page : 334 pages
File Size : 47,9 Mb
Release : 2008-10-29
Category : Mathematics
ISBN : 9783764387228
The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.
Author : Dmitri Burago,Yuri Burago,Sergei Ivanov
Publisher : American Mathematical Society
Page : 415 pages
File Size : 54,8 Mb
Release : 2022-01-27
Category : Mathematics
ISBN : 9781470468538
“Metric geometry” is an approach to geometry based on the notion of length on a topological space. This approach experienced a very fast development in the last few decades and penetrated into many other mathematical disciplines, such as group theory, dynamical systems, and partial differential equations. The objective of this graduate textbook is twofold: to give a detailed exposition of basic notions and techniques used in the theory of length spaces, and, more generally, to offer an elementary introduction into a broad variety of geometrical topics related to the notion of distance, including Riemannian and Carnot-Carathéodory metrics, the hyperbolic plane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (non-positively and non-negatively curved spaces). The authors tend to work with “easy-to-touch” mathematical objects using “easy-to-visualize” methods. The authors set a challenging goal of making the core parts of the book accessible to first-year graduate students. Most new concepts and methods are introduced and illustrated using simplest cases and avoiding technicalities. The book contains many exercises, which form a vital part of the exposition.
Author : Mikhail Gromov
Publisher : Springer Science & Business Media
Page : 594 pages
File Size : 52,5 Mb
Release : 2007-06-25
Category : Mathematics
ISBN : 9780817645830
This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book.
Author : Miroslav Bacak
Publisher : Walter de Gruyter GmbH & Co KG
Page : 217 pages
File Size : 50,9 Mb
Release : 2014-10-29
Category : Mathematics
ISBN : 9783110391084
In the past two decades, convex analysis and optimization have been developed in Hadamard spaces. This book represents a first attempt to give a systematic account on the subject. Hadamard spaces are complete geodesic spaces of nonpositive curvature. They include Hilbert spaces, Hadamard manifolds, Euclidean buildings and many other important spaces. While the role of Hadamard spaces in geometry and geometric group theory has been studied for a long time, first analytical results appeared as late as in the 1990s. Remarkably, it turns out that Hadamard spaces are appropriate for the theory of convex sets and convex functions outside of linear spaces. Since convexity underpins a large number of results in the geometry of Hadamard spaces, we believe that its systematic study is of substantial interest. Optimization methods then address various computational issues and provide us with approximation algorithms which may be useful in sciences and engineering. We present a detailed description of such an application to computational phylogenetics. The book is primarily aimed at both graduate students and researchers in analysis and optimization, but it is accessible to advanced undergraduate students as well.
Author : Miroslav Bacak
Publisher : Walter de Gruyter GmbH & Co KG
Page : 194 pages
File Size : 53,5 Mb
Release : 2014-10-29
Category : Mathematics
ISBN : 9783110361629
In the past two decades, convex analysis and optimization have been developed in Hadamard spaces. This book represents a first attempt to give a systematic account on the subject. Hadamard spaces are complete geodesic spaces of nonpositive curvature. They include Hilbert spaces, Hadamard manifolds, Euclidean buildings and many other important spaces. While the role of Hadamard spaces in geometry and geometric group theory has been studied for a long time, first analytical results appeared as late as in the 1990s. Remarkably, it turns out that Hadamard spaces are appropriate for the theory of convex sets and convex functions outside of linear spaces. Since convexity underpins a large number of results in the geometry of Hadamard spaces, we believe that its systematic study is of substantial interest. Optimization methods then address various computational issues and provide us with approximation algorithms which may be useful in sciences and engineering. We present a detailed description of such an application to computational phylogenetics. The book is primarily aimed at both graduate students and researchers in analysis and optimization, but it is accessible to advanced undergraduate students as well.
Author : Alberto Verjovski,Etienne Ghys,Andre Haefliger
Publisher : #N/A
Page : 744 pages
File Size : 48,6 Mb
Release : 1991-08-12
Category : Electronic
ISBN : 9789814569644
This proceedings presents the latest research materials done on group theory from geometrical viewpoint in particular Gromov's theory of hyperbolic groups, Coxeter groups, Tits buildings and actions on real trees. All these are very active subjects.
Author : Athanase Papadopoulos
Publisher : Springer Nature
Page : 396 pages
File Size : 42,5 Mb
Release : 2024-06-26
Category : Electronic
ISBN : 9783031435102
Author : Herbert Busemann
Publisher : Courier Corporation
Page : 434 pages
File Size : 52,5 Mb
Release : 2012-07-12
Category : Mathematics
ISBN : 9780486154626
A comprehensive approach to qualitative problems in intrinsic differential geometry, this text examines Desarguesian spaces, perpendiculars and parallels, covering spaces, the influence of the sign of the curvature on geodesics, more. 1955 edition. Includes 66 figures.
