A Course In Metric Geometry
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| Author | : Dmitri Burago |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 434 |
| Release | : 2001 |
| Genre | : Mathematics |
| ISBN | : 0821821296 |
"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-Caratheodory 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).
| Author | : Dmitri Burago |
| Publisher | : American Mathematical Society |
| Total Pages | : 415 |
| Release | : 2022-01-27 |
| Genre | : Mathematics |
| ISBN | : 1470468530 |
“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 | : Richard S. Millman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 394 |
| Release | : 1993-05-07 |
| Genre | : Mathematics |
| ISBN | : 9780387974125 |
Geometry: A Metric Approach with Models, imparts a real feeling for Euclidean and non-Euclidean (in particular, hyperbolic) geometry. Intended as a rigorous first course, the book introduces and develops the various axioms slowly, and then, in a departure from other texts, continually illustrates the major definitions and axioms with two or three models, enabling the reader to picture the idea more clearly. The second edition has been expanded to include a selection of expository exercises. Additionally, the authors have designed software with computational problems to accompany the text. This software may be obtained from George Parker.
| Author | : Stephanie Alexander |
| Publisher | : Springer |
| Total Pages | : 88 |
| Release | : 2019-05-08 |
| Genre | : Mathematics |
| ISBN | : 3030053121 |
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 | : Gerard Walschap |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 235 |
| Release | : 2012-08-23 |
| Genre | : Mathematics |
| ISBN | : 0387218262 |
This book offers an introduction to the theory of differentiable manifolds and fiber bundles. It examines bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres.
| Author | : Ernst Snapper |
| Publisher | : Elsevier |
| Total Pages | : 456 |
| Release | : 2014-05-10 |
| Genre | : Mathematics |
| ISBN | : 1483269337 |
Metric Affine Geometry focuses on linear algebra, which is the source for the axiom systems of all affine and projective geometries, both metric and nonmetric. This book is organized into three chapters. Chapter 1 discusses nonmetric affine geometry, while Chapter 2 reviews inner products of vector spaces. The metric affine geometry is treated in Chapter 3. This text specifically discusses the concrete model for affine space, dilations in terms of coordinates, parallelograms, and theorem of Desargues. The inner products in terms of coordinates and similarities of affine spaces are also elaborated. The prerequisites for this publication are a course in linear algebra and an elementary course in modern algebra that includes the concepts of group, normal subgroup, and quotient group. This monograph is suitable for students and aspiring geometry high school teachers.
| Author | : Martin R. Bridson |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 665 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 3662124947 |
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 | : S. Kumaresan |
| Publisher | : Springer |
| Total Pages | : 306 |
| Release | : 2002-01-15 |
| Genre | : Mathematics |
| ISBN | : 9386279088 |
| Author | : Paul J. Kelly |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2009 |
| Genre | : Convex bodies |
| ISBN | : 9780486469805 |
This text assumes no prerequisites, offering an easy-to-read treatment with simple notation and clear, complete proofs. From motivation to definition, its explanations feature concrete examples and theorems. 1979 edition.
| Author | : Gerald A. Edgar |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 252 |
| Release | : 2013-04-17 |
| Genre | : Mathematics |
| ISBN | : 1475741340 |
From the reviews: "In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. However, the book also contains many good illustrations of fractals (including 16 color plates), together with Logo programs which were used to generate them. ... Here then, at last, is an answer to the question on the lips of so many: 'What exactly is a fractal?' I do not expect many of this book's readers to achieve a mature understanding of this answer to the question, but anyone interested in finding out about the mathematics of fractal geometry could not choose a better place to start looking." #Mathematics Teaching#1
