Differential Geometry

Cartan's Generalization of Klein's Erlangen Program

Author: R.W. Sharpe

Publisher: Springer Science & Business Media

ISBN: 9780387947327

Category: Mathematics

Page: 421

View: 665

Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces giniralisis" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry.
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Riemannian Geometry and Geometric Analysis

Author: Jürgen Jost

Publisher: Springer

ISBN: 3319618601

Category: Mathematics

Page: 697

View: 7026

This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research. The 7th edition has been systematically reorganized and updated. Almost no page has been left unchanged. It also includes new material, for instance on symplectic geometry, as well as the Bishop-Gromov volume growth theorem which elucidates the geometric role of Ricci curvature. From the reviews:“This book provides a very readable introduction to Riemannian geometry and geometric analysis... With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome.” Mathematical Reviews “For readers familiar with the basics of differential geometry and some acquaintance with modern analysis, the book is reasonably self-contained. The book succeeds very well in laying out the foundations of modern Riemannian geometry and geometric analysis. It introduces a number of key techniques and provides a representative overview of the field.” Monatshefte für Mathematik
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Using the Mathematics Literature

Author: Kristine K. Fowler

Publisher: CRC Press

ISBN: 9780824750350

Category: Language Arts & Disciplines

Page: 475

View: 5580

This reference serves as a reader-friendly guide to every basic tool and skill required in the mathematical library and helps mathematicians find resources in any format in the mathematics literature. It lists a wide range of standard texts, journals, review articles, newsgroups, and Internet and database tools for every major subfield in mathematics and details methods of access to primary literature sources of new research, applications, results, and techniques. Using the Mathematics Literature is the most comprehensive and up-to-date resource on mathematics literature in both print and electronic formats, presenting time-saving strategies for retrieval of the latest information.
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Cartan for Beginners

Differential Geometry Via Moving Frames and Exterior Differential Systems

Author: Thomas Andrew Ivey,J. M. Landsberg

Publisher: American Mathematical Soc.

ISBN: 0821833758

Category: Mathematics

Page: 378

View: 413

This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems. It begins with the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics.One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. The book features an introduction to $G$-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.
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Principal Bundles

The Classical Case

Author: Stephen Bruce Sontz

Publisher: Springer

ISBN: 331914765X

Category: Science

Page: 280

View: 6502

This introductory graduate level text provides a relatively quick path to a special topic in classical differential geometry: principal bundles. While the topic of principal bundles in differential geometry has become classic, even standard, material in the modern graduate mathematics curriculum, the unique approach taken in this text presents the material in a way that is intuitive for both students of mathematics and of physics. The goal of this book is to present important, modern geometric ideas in a form readily accessible to students and researchers in both the physics and mathematics communities, providing each with an understanding and appreciation of the language and ideas of the other.
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Differential Geometry

Author: Heinrich W. Guggenheimer

Publisher: Courier Corporation

ISBN: 0486157202

Category: Mathematics

Page: 400

View: 6933

This text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; more. Includes exercises and 62 figures.
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What is Geometry?

Author: Giandomenico Sica

Publisher: Polimetrica s.a.s.

ISBN: 8876990305

Category: Mathematics

Page: 268

View: 6344

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Differential Geometry and Its Applications

Proceedings of the 10th International Conference, DGA 2007, Olomouc, Czech Republic, 27-31 August 2007

Author: N.A

Publisher: World Scientific

ISBN: 9812790616

Category: Electronic books

Page: 732

View: 9746

This volume contains invited lectures and selected research papers in the fields of classical and modern differential geometry, global analysis, and geometric methods in physics, presented at the 10th International Conference on Differential Geometry and its Applications (DGA2007), held in Olomouc, Czech Republic.The book covers recent developments and the latest results in the following fields: Riemannian geometry, connections, jets, differential invariants, the calculus of variations on manifolds, differential equations, Finsler structures, and geometric methods in physics. It is also a celebration of the 300th anniversary of the birth of one of the greatest mathematicians, Leonhard Euler, and includes the Euler lecture OC Leonhard Euler OCo 300 years onOCO by R Wilson. Notable contributors include J F Cariena, M Castrilln Lpez, J Erichhorn, J-H Eschenburg, I KoliO, A P Kopylov, J Korbai, O Kowalski, B Kruglikov, D Krupka, O Krupkovi, R L(r)andre, Haizhong Li, S Maeda, M A Malakhaltsev, O I Mokhov, J Muoz Masqu(r), S Preston, V Rovenski, D J Saunders, M Sekizawa, J Slovik, J Szilasi, L Tamissy, P Walczak, and others."
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An Alternative Approach to Lie Groups and Geometric Structures

Author: Ercüment Ortaçgil

Publisher: Oxford University Press

ISBN: 0192554840

Category: Mathematics

Page: 240

View: 2961

This book presents a new and innovative approach to Lie groups and differential geometry. Rather than compiling and reviewing the existing material on this classical subject, Professor Ortaçgil instead questions the foundations of the subject, and proposes a new direction. Aimed at the curious and courageous mathematician, this book aims to provoke further debate and inspire further development of this original research.
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Differential Manifolds

Author: Antoni A. Kosinski

Publisher: Academic Press

ISBN: 9780080874586

Category: Mathematics

Page: 248

View: 6888

Differential Manifolds is a modern graduate-level introduction to the important field of differential topology. The concepts of differential topology lie at the heart of many mathematical disciplines such as differential geometry and the theory of lie groups. The book introduces both the h-cobordism theorem and the classification of differential structures on spheres. The presentation of a number of topics in a clear and simple fashion make this book an outstanding choice for a graduate course in differential topology as well as for individual study. Presents the study and classification of smooth structures on manifolds It begins with the elements of theory and concludes with an introduction to the method of surgery Chapters 1-5 contain a detailed presentation of the foundations of differential topology--no knowledge of algebraic topology is required for this self-contained section Chapters 6-8 begin by explaining the joining of manifolds along submanifolds, and ends with the proof of the h-cobordism theory Chapter 9 presents the Pontriagrin construction, the principle link between differential topology and homotopy theory; The final chapter introduces the method of surgery and applies it to the classification of smooth structures on spheres
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Riemann, Topology, and Physics

Author: Michael I. Monastyrsky

Publisher: Springer Science & Business Media

ISBN: 9780817647780

Category: Mathematics

Page: 215

View: 9692

The significantly expanded second edition of this book combines a fascinating account of the life and work of Bernhard Riemann with a lucid discussion of current interaction between topology and physics. The author, a distinguished mathematical physicist, takes into account his own research at the Riemann archives of Göttingen University and developments over the last decade that connect Riemann with numerous significant ideas and methods reflected throughout contemporary mathematics and physics. Special attention is paid in part one to results on the Riemann–Hilbert problem and, in part two, to discoveries in field theory and condensed matter.
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Diffeology

Author: Patrick Iglesias-Zemmour

Publisher: American Mathematical Soc.

ISBN: 0821891316

Category: Mathematics

Page: 439

View: 6223

Diffeology is an extension of differential geometry. With a minimal set of axioms, diffeology allows us to deal simply but rigorously with objects which do not fall within the usual field of differential geometry: quotients of manifolds (even non-Hausdorff), spaces of functions, groups of diffeomorphisms, etc. The category of diffeology objects is stable under standard set-theoretic operations, such as quotients, products, co-products, subsets, limits, and co-limits. With its right balance between rigor and simplicity, diffeology can be a good framework for many problems that appear in various areas of physics. Actually, the book lays the foundations of the main fields of differential geometry used in theoretical physics: differentiability, Cartan differential calculus, homology and cohomology, diffeological groups, fiber bundles, and connections. The book ends with an open program on symplectic diffeology, a rich field of application of the theory. Many exercises with solutions make this book appropriate for learning the subject.
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Supersymmetry for Mathematicians

An Introduction

Author: V. S. Varadarajan

Publisher: American Mathematical Soc.

ISBN: 0821835742

Category: Mathematics

Page: 300

View: 3982

Supersymmetry has been the object of study by theoretical physicists since the early 1970's. In recent years it has attracted the interest of mathematicians because of its novelty, and because of significance, both in mathematics and physics, of the main issues it raises. This book presents the foundations of supersymmetry to the mathematically minded reader in a cogent and self-contained manner. It begins with a brief introduction to the physical foundations of the theory, especially the classification of relativistic particles and their wave equations, such as the equations of Dirac and Weyl. It then continues the development of the theory of supermanifolds stressing the analogy with the Grothendieck theory of schemes. All the super linear algebra needed for the book is developed here and the basic theorems are established: differential and integral calculus in supermanifolds, Frobenius theorem, foundations of the theory of super Lie groups, and so on. A special feature of the book is the treatment in depth of the theory of spinors in all dimensions and signatures, which is the basis of all developments of supergeometry both in physics and mathematics, especially in quantum field theory and supergravity.
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Conformal Differential Geometry and Its Generalizations

Author: Maks A. Akivis,Maks Aĭzikovich Akivis,Vladislav V. Goldberg,Vladislav Viktorovich Goldberg

Publisher: John Wiley & Sons

ISBN: 9780471149583

Category: Mathematics

Page: 383

View: 6408

Comprehensive coverage of the foundations, applications, recentdevelopments, and future of conformal differential geometry Conformal Differential Geometry and Its Generalizations is thefirst and only text that systematically presents the foundationsand manifestations of conformal differential geometry. It offersthe first unified presentation of the subject, which wasestablished more than a century ago. The text is divided into sevenchapters, each containing figures, formulas, and historical andbibliographical notes, while numerous examples elucidate thenecessary theory. Clear, focused, and expertly synthesized, Conformal DifferentialGeometry and Its Generalizations * Develops the theory of hypersurfaces and submanifolds of anydimension of conformal and pseudoconformal spaces. * Investigates conformal and pseudoconformal structures on amanifold of arbitrary dimension, derives their structure equations,and explores their tensor of conformal curvature. * Analyzes the real theory of four-dimensional conformal structuresof all possible signatures. * Considers the analytic and differential geometry of Grassmann andalmost Grassmann structures. * Draws connections between almost Grassmann structures and webtheory. Conformal differential geometry, a part of classical differentialgeometry, was founded at the turn of the century and gave rise tothe study of conformal and almost Grassmann structures in lateryears. Until now, no book has offered a systematic presentation ofthe multidimensional conformal differential geometry and theconformal and almost Grassmann structures. After years of intense research at their respective universitiesand at the Soviet School of Differential Geometry, Maks A. Akivisand Vladislav V. Goldberg have written this well-conceived,expertly executed volume to fill a void in the literature. Dr.Akivis and Dr. Goldberg supply a deep foundation, applications,numerous examples, and recent developments in the field. Many ofthe findings that fill these pages are published here for the firsttime, and previously published results are reexamined in a unifiedcontext. The geometry and theory of conformal and pseudoconformal spaces ofarbitrary dimension, as well as the theory of Grassmann and almostGrassmann structures, are discussed and analyzed in detail. Thetopics covered not only advance the subject itself, but poseimportant questions for future investigations. This exhaustive,groundbreaking text combines the classical results and recentdevelopments and findings. This volume is intended for graduate students and researchers ofdifferential geometry. It can be especially useful to thosestudents and researchers who are interested in conformal andGrassmann differential geometry and their applications totheoretical physics.
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An Introduction to Dynamical Systems and Chaos

Author: G.C. Layek

Publisher: Springer

ISBN: 8132225562

Category: Mathematics

Page: 622

View: 736

The book discusses continuous and discrete systems in systematic and sequential approaches for all aspects of nonlinear dynamics. The unique feature of the book is its mathematical theories on flow bifurcations, oscillatory solutions, symmetry analysis of nonlinear systems and chaos theory. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, and a number of examples worked out in detail and exercises have been included. Chapters 1–8 are devoted to continuous systems, beginning with one-dimensional flows. Symmetry is an inherent character of nonlinear systems, and the Lie invariance principle and its algorithm for finding symmetries of a system are discussed in Chap. 8. Chapters 9–13 focus on discrete systems, chaos and fractals. Conjugacy relationship among maps and its properties are described with proofs. Chaos theory and its connection with fractals, Hamiltonian flows and symmetries of nonlinear systems are among the main focuses of this book. Over the past few decades, there has been an unprecedented interest and advances in nonlinear systems, chaos theory and fractals, which is reflected in undergraduate and postgraduate curricula around the world. The book is useful for courses in dynamical systems and chaos, nonlinear dynamics, etc., for advanced undergraduate and postgraduate students in mathematics, physics and engineering.
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Elementary Differential Geometry

Author: A.N. Pressley

Publisher: Springer Science & Business Media

ISBN: 1848828918

Category: Mathematics

Page: 474

View: 6258

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include: a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature. Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com ul>
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Theory of Complex Functions

Author: Reinhold Remmert

Publisher: Springer Science & Business Media

ISBN: 1461209390

Category: Mathematics

Page: 458

View: 4866

A lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. There is also ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations - in the original language with their English translation - from their classical works. Yet the book is far from being a mere history of function theory, and even experts will find a few new or long forgotten gems here. Destined to accompany students making their way into this classical area of mathematics, the book offers quick access to the essential results for exam preparation. Teachers and interested mathematicians in finance, industry and science will profit from reading this again and again, and will refer back to it with pleasure.
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Dynamical Systems and Chaos

Author: Henk Broer,Floris Takens

Publisher: Springer Science & Business Media

ISBN: 9781441968708

Category: Mathematics

Page: 313

View: 7345

Over the last four decades there has been extensive development in the theory of dynamical systems. This book aims at a wide audience where the first four chapters have been used for an undergraduate course in Dynamical Systems. Material from the last two chapters and from the appendices has been used quite a lot for master and PhD courses. All chapters are concluded by an exercise section. The book is also directed towards researchers, where one of the challenges is to help applied researchers acquire background for a better understanding of the data that computer simulation or experiment may provide them with the development of the theory.
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