Cartan for Beginners

Cartan for Beginners

Author: Thomas Andrew Ivey

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 394

ISBN-13: 0821833758

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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.


Book Synopsis Cartan for Beginners by : Thomas Andrew Ivey

Download or read book Cartan for Beginners written by Thomas Andrew Ivey and published by American Mathematical Soc.. This book was released on 2003 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Cartan for Beginners

Cartan for Beginners

Author: Thomas A. Ivey

Publisher: American Mathematical Soc.

Published: 2016-12-15

Total Pages: 455

ISBN-13: 1470409860

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Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) 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 in geometry. It begins with the classical differential 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. As well, the book features an introduction to G-structures and a treatment of the theory of connections. The techniques of EDS are 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 geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields. The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.


Book Synopsis Cartan for Beginners by : Thomas A. Ivey

Download or read book Cartan for Beginners written by Thomas A. Ivey and published by American Mathematical Soc.. This book was released on 2016-12-15 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) 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 in geometry. It begins with the classical differential 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. As well, the book features an introduction to G-structures and a treatment of the theory of connections. The techniques of EDS are 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 geometry of PDE systems and complex algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields. The second edition features three new chapters: on Riemannian geometry, emphasizing the use of representation theory; on the latest developments in the study of Darboux-integrable systems; and on conformal geometry, written in a manner to introduce readers to the related parabolic geometry perspective.

From Frenet to Cartan: The Method of Moving Frames

From Frenet to Cartan: The Method of Moving Frames

Author: Jeanne N. Clelland

Publisher: American Mathematical Soc.

Published: 2017-03-29

Total Pages: 414

ISBN-13: 1470429527

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The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others. This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames on Riemannian manifolds. The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed guidance regarding the use of the computer algebra system Maple™ to perform many of the computations involved in the exercises.


Book Synopsis From Frenet to Cartan: The Method of Moving Frames by : Jeanne N. Clelland

Download or read book From Frenet to Cartan: The Method of Moving Frames written by Jeanne N. Clelland and published by American Mathematical Soc.. This book was released on 2017-03-29 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others. This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames on Riemannian manifolds. The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed guidance regarding the use of the computer algebra system Maple™ to perform many of the computations involved in the exercises.

Hadamard's Plane Geometry

Hadamard's Plane Geometry

Author: Mark E. Saul

Publisher: American Mathematical Soc.

Published: 2010-02-10

Total Pages: 362

ISBN-13: 0821843680

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Jacques Hadamard, among the greatest mathematicians of the twentieth century, made signal contributions to a number of fields. But his mind could not be confined to the upper reaches of mathematical thought. He also produced a massive two-volume work, on plane and solid geometry, for pre-college teachers in the French school system. In those books, Hadamard's style invites participation. His exposition is minimal, providing only the results necessary to support the solution of the many elegant problems he poses afterwards. That is, the problems interpret the text in the way that harmony interprets melody in a well-composed piece of music. The present volume offers solutions to the problems in the first part of Hadamard's work (Lessons in Geometry. I. Plane Geometry, Jacques Hadamard, Amer. Math. Soc. (2008)), and can be viewed as a reader's companion to that book. It requires of the reader only the background of high school plane geometry, which Lessons in Geometry provides. The solutions strive to connect the general methods given in the text with intuitions that are natural to the subject, giving as much motivation as possible as well as rigorous and formal solutions. Ideas for further exploration are often suggested, as well as hints for classroom use. This book will be of interest to high school teachers, gifted high school students, college students, and those mathematics majors interested in geometry.


Book Synopsis Hadamard's Plane Geometry by : Mark E. Saul

Download or read book Hadamard's Plane Geometry written by Mark E. Saul and published by American Mathematical Soc.. This book was released on 2010-02-10 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Jacques Hadamard, among the greatest mathematicians of the twentieth century, made signal contributions to a number of fields. But his mind could not be confined to the upper reaches of mathematical thought. He also produced a massive two-volume work, on plane and solid geometry, for pre-college teachers in the French school system. In those books, Hadamard's style invites participation. His exposition is minimal, providing only the results necessary to support the solution of the many elegant problems he poses afterwards. That is, the problems interpret the text in the way that harmony interprets melody in a well-composed piece of music. The present volume offers solutions to the problems in the first part of Hadamard's work (Lessons in Geometry. I. Plane Geometry, Jacques Hadamard, Amer. Math. Soc. (2008)), and can be viewed as a reader's companion to that book. It requires of the reader only the background of high school plane geometry, which Lessons in Geometry provides. The solutions strive to connect the general methods given in the text with intuitions that are natural to the subject, giving as much motivation as possible as well as rigorous and formal solutions. Ideas for further exploration are often suggested, as well as hints for classroom use. This book will be of interest to high school teachers, gifted high school students, college students, and those mathematics majors interested in geometry.

Exterior Differential Systems

Exterior Differential Systems

Author: Robert L. Bryant

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 483

ISBN-13: 1461397146

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This book gives a treatment of exterior differential systems. It will in clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of inde pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepen dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object.


Book Synopsis Exterior Differential Systems by : Robert L. Bryant

Download or read book Exterior Differential Systems written by Robert L. Bryant and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 483 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a treatment of exterior differential systems. It will in clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of inde pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepen dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object.

Complex Geometry

Complex Geometry

Author: Daniel Huybrechts

Publisher: Springer Science & Business Media

Published: 2005

Total Pages: 336

ISBN-13: 9783540212904

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Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)


Book Synopsis Complex Geometry by : Daniel Huybrechts

Download or read book Complex Geometry written by Daniel Huybrechts and published by Springer Science & Business Media. This book was released on 2005 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)

The Road to Reality

The Road to Reality

Author: Roger Penrose

Publisher: Vintage

Published: 2021-06-09

Total Pages: 1136

ISBN-13: 0593315308

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**WINNER OF THE 2020 NOBEL PRIZE IN PHYSICS** The Road to Reality is the most important and ambitious work of science for a generation. It provides nothing less than a comprehensive account of the physical universe and the essentials of its underlying mathematical theory. It assumes no particular specialist knowledge on the part of the reader, so that, for example, the early chapters give us the vital mathematical background to the physical theories explored later in the book. Roger Penrose's purpose is to describe as clearly as possible our present understanding of the universe and to convey a feeling for its deep beauty and philosophical implications, as well as its intricate logical interconnections. The Road to Reality is rarely less than challenging, but the book is leavened by vivid descriptive passages, as well as hundreds of hand-drawn diagrams. In a single work of colossal scope one of the world's greatest scientists has given us a complete and unrivalled guide to the glories of the universe that we all inhabit. 'Roger Penrose is the most important physicist to work in relativity theory except for Einstein. He is one of the very few people I've met in my life who, without reservation, I call a genius' Lee Smolin


Book Synopsis The Road to Reality by : Roger Penrose

Download or read book The Road to Reality written by Roger Penrose and published by Vintage. This book was released on 2021-06-09 with total page 1136 pages. Available in PDF, EPUB and Kindle. Book excerpt: **WINNER OF THE 2020 NOBEL PRIZE IN PHYSICS** The Road to Reality is the most important and ambitious work of science for a generation. It provides nothing less than a comprehensive account of the physical universe and the essentials of its underlying mathematical theory. It assumes no particular specialist knowledge on the part of the reader, so that, for example, the early chapters give us the vital mathematical background to the physical theories explored later in the book. Roger Penrose's purpose is to describe as clearly as possible our present understanding of the universe and to convey a feeling for its deep beauty and philosophical implications, as well as its intricate logical interconnections. The Road to Reality is rarely less than challenging, but the book is leavened by vivid descriptive passages, as well as hundreds of hand-drawn diagrams. In a single work of colossal scope one of the world's greatest scientists has given us a complete and unrivalled guide to the glories of the universe that we all inhabit. 'Roger Penrose is the most important physicist to work in relativity theory except for Einstein. He is one of the very few people I've met in my life who, without reservation, I call a genius' Lee Smolin

Integrated Reservoir Stu...

Integrated Reservoir Stu...

Author: Luca Consentino

Publisher: Editions OPHRYS

Published: 2001

Total Pages: 344

ISBN-13: 9782710811176

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Annotation The goal of this book is to highlight the difference between an integrated reservoir study and a traditional one. The benefits of integrated studies are outlined, and consider its implications for everyday working conditions. Technical and professional challenges are discussed and necessary changes are detailed, with emphasis on the role of the project leader. Chapters consider elements like the integrated database, the integrated geological model, rock properties, hydrocarbon in place determination, reservoir engineering, numerical reservoir simulation, and planning for a study. Cosentino is a reservoir engineer and project manager for a private firm. c. Book News Inc.


Book Synopsis Integrated Reservoir Stu... by : Luca Consentino

Download or read book Integrated Reservoir Stu... written by Luca Consentino and published by Editions OPHRYS. This book was released on 2001 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: Annotation The goal of this book is to highlight the difference between an integrated reservoir study and a traditional one. The benefits of integrated studies are outlined, and consider its implications for everyday working conditions. Technical and professional challenges are discussed and necessary changes are detailed, with emphasis on the role of the project leader. Chapters consider elements like the integrated database, the integrated geological model, rock properties, hydrocarbon in place determination, reservoir engineering, numerical reservoir simulation, and planning for a study. Cosentino is a reservoir engineer and project manager for a private firm. c. Book News Inc.

Differential Geometry for Physicists and Mathematicians

Differential Geometry for Physicists and Mathematicians

Author: José G. Vargas

Publisher: World Scientific Publishing Company Incorporated

Published: 2014

Total Pages: 293

ISBN-13: 9789814566391

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I. Introduction. 1. Orientations -- II. Tools. 2. Differential forms -- 3. Vector spaces and tensor products -- 4. Exterior differentiation -- III. Two Klein geometries. 5. Affine Klein geometry -- 6. Euclidean Klein geometry -- IV. Cartan connections. 7. Generalized geometry made simple -- 8. Affine connections -- 9. Euclidean connections -- 10. Riemannian spaces and pseudo-spaces -- V. The future? 11. Extensions of Cartan -- 12. Understand the past to imagine the future -- 13. A book of farewells


Book Synopsis Differential Geometry for Physicists and Mathematicians by : José G. Vargas

Download or read book Differential Geometry for Physicists and Mathematicians written by José G. Vargas and published by World Scientific Publishing Company Incorporated. This book was released on 2014 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: I. Introduction. 1. Orientations -- II. Tools. 2. Differential forms -- 3. Vector spaces and tensor products -- 4. Exterior differentiation -- III. Two Klein geometries. 5. Affine Klein geometry -- 6. Euclidean Klein geometry -- IV. Cartan connections. 7. Generalized geometry made simple -- 8. Affine connections -- 9. Euclidean connections -- 10. Riemannian spaces and pseudo-spaces -- V. The future? 11. Extensions of Cartan -- 12. Understand the past to imagine the future -- 13. A book of farewells

Introduction to Lie Algebras

Introduction to Lie Algebras

Author: K. Erdmann

Publisher: Springer Science & Business Media

Published: 2006-09-28

Total Pages: 254

ISBN-13: 1846284902

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Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.


Book Synopsis Introduction to Lie Algebras by : K. Erdmann

Download or read book Introduction to Lie Algebras written by K. Erdmann and published by Springer Science & Business Media. This book was released on 2006-09-28 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.