Metric Spaces Convexity And Nonpositive Curvature PDF Download
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Author | : Athanase Papadopoulos |
Publisher | : European Mathematical Society |
Total Pages | : 306 |
Release | : 2005 |
Genre | : Computers |
ISBN | : 9783037190104 |
Download Metric Spaces, Convexity and Nonpositive Curvature Book in PDF, ePub and Kindle
Author | : Martin R. Bridson |
Publisher | : Springer Science & Business Media |
Total Pages | : 680 |
Release | : 2011-10-20 |
Genre | : Mathematics |
ISBN | : 9783540643241 |
Download Metric Spaces of Non-Positive Curvature Book in PDF, ePub and Kindle
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 | : Springer Science & Business Media |
Total Pages | : 124 |
Release | : 1997-05-01 |
Genre | : Mathematics |
ISBN | : 9783764357368 |
Download Nonpositive Curvature: Geometric and Analytic Aspects Book in PDF, ePub and Kindle
The present book contains the lecture notes from a "Nachdiplomvorlesung", a topics course adressed to Ph. D. students, at the ETH ZUrich during the winter term 95/96. Consequently, these notes are arranged according to the requirements of organizing the material for oral exposition, and the level of difficulty and the exposition were adjusted to the audience in Zurich. The aim of the course was to introduce some geometric and analytic concepts that have been found useful in advancing our understanding of spaces of nonpos itive curvature. In particular in recent years, it has been realized that often it is useful for a systematic understanding not to restrict the attention to Riemannian manifolds only, but to consider more general classes of metric spaces of generalized nonpositive curvature. The basic idea is to isolate a property that on one hand can be formulated solely in terms of the distance function and on the other hand is characteristic of nonpositive sectional curvature on a Riemannian manifold, and then to take this property as an axiom for defining a metric space of nonposi tive curvature. Such constructions have been put forward by Wald, Alexandrov, Busemann, and others, and they will be systematically explored in Chapter 2. Our focus and treatment will often be different from the existing literature. In the first Chapter, we consider several classes of examples of Riemannian manifolds of nonpositive curvature, and we explain how conditions about nonpos itivity or negativity of curvature can be exploited in various geometric contexts.
Author | : Werner Ballmann |
Publisher | : Birkhäuser |
Total Pages | : 114 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3034892403 |
Download Lectures on Spaces of Nonpositive Curvature Book in PDF, ePub and Kindle
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 | : Werner Ballmann |
Publisher | : Springer Science & Business Media |
Total Pages | : 280 |
Release | : 2013-12-11 |
Genre | : Mathematics |
ISBN | : 1468491598 |
Download Manifolds of Nonpositive Curvature Book in PDF, ePub and Kindle
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 | : Luigi Ambrosio |
Publisher | : Springer Science & Business Media |
Total Pages | : 333 |
Release | : 2008-10-29 |
Genre | : Mathematics |
ISBN | : 376438722X |
Download Gradient Flows Book in PDF, ePub and Kindle
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 | : Miroslav Bačák |
Publisher | : |
Total Pages | : |
Release | : 2014 |
Genre | : |
ISBN | : |
Download A Notion of Nonpositive Curvature for General Metric Spaces Book in PDF, ePub and Kindle
Author | : Stephanie Alexander |
Publisher | : Springer |
Total Pages | : 88 |
Release | : 2019-05-08 |
Genre | : Mathematics |
ISBN | : 3030053121 |
Download An Invitation to Alexandrov Geometry Book in PDF, ePub and Kindle
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 | : Miroslav Bacak |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 217 |
Release | : 2014-10-29 |
Genre | : Mathematics |
ISBN | : 3110391082 |
Download Convex Analysis and Optimization in Hadamard Spaces Book in PDF, ePub and Kindle
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 | : Werner Ballmann |
Publisher | : Birkhauser |
Total Pages | : 112 |
Release | : 1995 |
Genre | : Mathematics |
ISBN | : 9780817652425 |
Download Lectures on Spaces of Nonpositive Curvature Book in PDF, ePub and Kindle
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. With a few exceptions, the book is self-contained and can be used as a text for a seminar or a reading course. Some acquaintance with basic notions and techniques from Riemannian geometry is helpful, in particular for Chapter IV.