A Spinorial Approach To Riemannian And Conformal Geometry

Author : Jean-Pierre Bourguignon,Oussama Hijazi,Jean-Louis Milhorat,Andrei Moroianu,Sergiu Moroianu
Publisher : Erich Schmidt Verlag GmbH & Co. KG
Page : 468 pages
File Size : 48,8 Mb
Release : 2015
Category : Clifford algebras
ISBN : 3037191368

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A Spinorial Approach to Riemannian and Conformal Geometry by Jean-Pierre Bourguignon,Oussama Hijazi,Jean-Louis Milhorat,Andrei Moroianu,Sergiu Moroianu Pdf

The book gives an elementary and comprehensive introduction to Spin Geometry, with particular emphasis on the Dirac operator, which plays a fundamental role in differential geometry and mathematical physics. After a self-contained presentation of the basic algebraic, geometrical, analytical and topological ingredients, a systematic study of the spectral properties of the Dirac operator on compact spin manifolds is carried out. The classical estimates on eigenvalues and their limiting cases are discussed next, highlighting the subtle interplay of spinors and special geometric structures. Several applications of these ideas are presented, including spinorial proofs of the Positive Mass Theorem or the classification of positive Kahler-Einstein contact manifolds. Representation theory is used to explicitly compute the Dirac spectrum of compact symmetric spaces. The special features of the book include a unified treatment of $\mathrm{Spin^c}$ and conformal spin geometry (with special emphasis on the conformal covariance of the Dirac operator), an overview with proofs of the theory of elliptic differential operators on compact manifolds based on pseudodifferential calculus, a spinorial characterization of special geometries, and a self-contained presentation of the representation-theoretical tools needed in order to apprehend spinors. This book will help advanced graduate students and researchers to get more familiar with this beautiful, though not sufficiently known, domain of mathematics with great relevance to both theoretical physics and geometry.

Author : Pierre Anglès
Publisher : Birkhäuser
Page : 0 pages
File Size : 55,8 Mb
Release : 2008-11-01
Category : Mathematics
ISBN : 0817670440

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Conformal Groups in Geometry and Spin Structures by Pierre Anglès Pdf

This book provides a self-contained overview of the role of conformal groups in geometry and mathematical physics. It features a careful development of the material, from the basics of Clifford algebras to more advanced topics. Each chapter covers a specific aspect of conformal groups and conformal spin geometry. All major concepts are introduced and followed by detailed descriptions and definitions, and a comprehensive bibliography and index round out the work. Rich in exercises that are accompanied by full proofs and many hints, the book will be ideal as a course text or self-study volume for senior undergraduates and graduate students.

Author : Lutz Habermann
Publisher : Springer
Page : 123 pages
File Size : 47,7 Mb
Release : 2007-05-06
Category : Mathematics
ISBN : 9783540444435

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Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures by Lutz Habermann Pdf

This monograph deals with recent questions of conformal geometry. It provides in detail an approach to studying moduli spaces of conformal structures, using a new canonical metric for conformal structures. This book is accessible to readers with basic knowledge in differential geometry and global analysis. It addresses graduates and researchers.

Author : Steven G. Krantz
Publisher : Courier Dover Publications
Page : 304 pages
File Size : 42,6 Mb
Release : 2016-03-17
Category : Mathematics
ISBN : 9780486810324

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The Theory and Practice of Conformal Geometry by Steven G. Krantz Pdf

In this original text, prolific mathematics author Steven G. Krantz addresses conformal geometry, a subject that has occupied him for four decades and for which he helped to develop some of the modern theory. This book takes readers with a basic grounding in complex variable theory to the forefront of some of the current approaches to the topic. "Along the way," the author notes in his Preface, "the reader will be exposed to some beautiful function theory and also some of the rudiments of geometry and analysis that make this subject so vibrant and lively." More up-to-date and accessible to advanced undergraduates than most of the other books available in this specific field, the treatment discusses the history of this active and popular branch of mathematics as well as recent developments. Topics include the Riemann mapping theorem, invariant metrics, normal families, automorphism groups, the Schwarz lemma, harmonic measure, extremal length, analytic capacity, and invariant geometry. A helpful Bibliography and Index complete the text.

Author : Jürgen Jost
Publisher : Springer
Page : 697 pages
File Size : 51,7 Mb
Release : 2017-10-13
Category : Mathematics
ISBN : 9783319618609

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Riemannian Geometry and Geometric Analysis by Jürgen Jost Pdf

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

Author : Sebastià Xambó-Descamps
Publisher : Springer
Page : 151 pages
File Size : 47,5 Mb
Release : 2018-11-22
Category : Mathematics
ISBN : 9783030004040

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Real Spinorial Groups by Sebastià Xambó-Descamps Pdf

This book explores the Lipschitz spinorial groups (versor, pinor, spinor and rotor groups) of a real non-degenerate orthogonal geometry (or orthogonal geometry, for short) and how they relate to the group of isometries of that geometry. After a concise mathematical introduction, it offers an axiomatic presentation of the geometric algebra of an orthogonal geometry. Once it has established the language of geometric algebra (linear grading of the algebra; geometric, exterior and interior products; involutions), it defines the spinorial groups, demonstrates their relation to the isometry groups, and illustrates their suppleness (geometric covariance) with a variety of examples. Lastly, the book provides pointers to major applications, an extensive bibliography and an alphabetic index. Combining the characteristics of a self-contained research monograph and a state-of-the-art survey, this book is a valuable foundation reference resource on applications for both undergraduate and graduate students.

Author : Xinyue Cheng,Zhongmin Shen
Publisher : Springer Science & Business Media
Page : 149 pages
File Size : 43,7 Mb
Release : 2013-01-29
Category : Mathematics
ISBN : 9783642248887

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Finsler Geometry by Xinyue Cheng,Zhongmin Shen Pdf

"Finsler Geometry: An Approach via Randers Spaces" exclusively deals with a special class of Finsler metrics -- Randers metrics, which are defined as the sum of a Riemannian metric and a 1-form. Randers metrics derive from the research on General Relativity Theory and have been applied in many areas of the natural sciences. They can also be naturally deduced as the solution of the Zermelo navigation problem. The book provides readers not only with essential findings on Randers metrics but also the core ideas and methods which are useful in Finsler geometry. It will be of significant interest to researchers and practitioners working in Finsler geometry, even in differential geometry or related natural fields. Xinyue Cheng is a Professor at the School of Mathematics and Statistics of Chongqing University of Technology, China. Zhongmin Shen is a Professor at the Department of Mathematical Sciences of Indiana University Purdue University, USA.

Author : Nicolas Ginoux
Publisher : Springer
Page : 156 pages
File Size : 50,7 Mb
Release : 2009-05-30
Category : Mathematics
ISBN : 9783642015700

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The Dirac Spectrum by Nicolas Ginoux Pdf

This volume surveys the spectral properties of the spin Dirac operator. After a brief introduction to spin geometry, it presents the main known estimates for Dirac eigenvalues on compact manifolds with or without boundaries.

Author : Sun-Yung A. Chang,Paul C. Yang,Karsten Grove,Jon G. Wolfson
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 46,6 Mb
Release : 2002
Category : Conformal geometry
ISBN : 9780821832103

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Conformal, Riemannian and Lagrangian Geometry by Sun-Yung A. Chang,Paul C. Yang,Karsten Grove,Jon G. Wolfson Pdf

Recent developments in topology and analysis have led to the creation of new lines of investigation in differential geometry. The 2000 Barrett Lectures present the background, context and main techniques of three such lines by means of surveys by leading researchers. The first chapter (by Alice Chang and Paul Yang) introduces new classes of conformal geometric invariants, and then applies powerful techniques in nonlinear differential equations to derive results on compactificationsof manifolds and on Yamabe-type variational problems for these invariants. This is followed by Karsten Grove's lectures, which focus on the use of isometric group actions and metric geometry techniques to understand new examples and classification results in Riemannian geometry, especially inconnection with positive curvature. The chapter written by Jon Wolfson introduces the emerging field of Lagrangian variational problems, which blends in novel ways the structures of symplectic geometry and the techniques of the modern calculus of variations. The lectures provide an up-do-date overview and an introduction to the research literature in each of their areas. The book is a very enjoyable read, which should prove useful to graduate students and researchers in differential geometryand geometric analysis.

Author : Jean-Michel Morel,Bernard Teissier
Publisher : Springer Nature
Page : 629 pages
File Size : 54,5 Mb
Release : 2023-06-14
Category : Mathematics
ISBN : 9783031122446

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Mathematics Going Forward by Jean-Michel Morel,Bernard Teissier Pdf

This volume is an original collection of articles by 44 leading mathematicians on the theme of the future of the discipline. The contributions range from musings on the future of specific fields, to analyses of the history of the discipline, to discussions of open problems and conjectures, including first solutions of unresolved problems. Interestingly, the topics do not cover all of mathematics, but only those deemed most worthy to reflect on for future generations. These topics encompass the most active parts of pure and applied mathematics, including algebraic geometry, probability, logic, optimization, finance, topology, partial differential equations, category theory, number theory, differential geometry, dynamical systems, artificial intelligence, theory of groups, mathematical physics and statistics.

Author : Francis E. Burstall,Dirk Ferus,Katrin Leschke,Franz Pedit,Ulrich Pinkall
Publisher : Springer
Page : 96 pages
File Size : 48,7 Mb
Release : 2004-10-20
Category : Mathematics
ISBN : 9783540453017

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Conformal Geometry of Surfaces in S4 and Quaternions by Francis E. Burstall,Dirk Ferus,Katrin Leschke,Franz Pedit,Ulrich Pinkall Pdf

The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bäcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.

Author : Maks A. Akivis,Vladislav V. Goldberg
Publisher : John Wiley & Sons
Page : 404 pages
File Size : 43,5 Mb
Release : 2011-09-20
Category : Mathematics
ISBN : 9781118030882

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Conformal Differential Geometry and Its Generalizations by Maks A. Akivis,Vladislav V. Goldberg Pdf

Comprehensive coverage of the foundations, applications, recent developments, and future of conformal differential geometry Conformal Differential Geometry and Its Generalizations is the first and only text that systematically presents the foundations and manifestations of conformal differential geometry. It offers the first unified presentation of the subject, which was established more than a century ago. The text is divided into seven chapters, each containing figures, formulas, and historical and bibliographical notes, while numerous examples elucidate the necessary theory. Clear, focused, and expertly synthesized, Conformal Differential Geometry and Its Generalizations * Develops the theory of hypersurfaces and submanifolds of any dimension of conformal and pseudoconformal spaces. * Investigates conformal and pseudoconformal structures on a manifold of arbitrary dimension, derives their structure equations, and explores their tensor of conformal curvature. * Analyzes the real theory of four-dimensional conformal structures of all possible signatures. * Considers the analytic and differential geometry of Grassmann and almost Grassmann structures. * Draws connections between almost Grassmann structures and web theory. Conformal differential geometry, a part of classical differential geometry, was founded at the turn of the century and gave rise to the study of conformal and almost Grassmann structures in later years. Until now, no book has offered a systematic presentation of the multidimensional conformal differential geometry and the conformal and almost Grassmann structures. After years of intense research at their respective universities and at the Soviet School of Differential Geometry, Maks A. Akivis and 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 of the findings that fill these pages are published here for the first time, and previously published results are reexamined in a unified context. The geometry and theory of conformal and pseudoconformal spaces of arbitrary dimension, as well as the theory of Grassmann and almost Grassmann structures, are discussed and analyzed in detail. The topics covered not only advance the subject itself, but pose important questions for future investigations. This exhaustive, groundbreaking text combines the classical results and recent developments and findings. This volume is intended for graduate students and researchers of differential geometry. It can be especially useful to those students and researchers who are interested in conformal and Grassmann differential geometry and their applications to theoretical physics.

Author : Norbert A'Campo
Publisher : Springer Nature
Page : 282 pages
File Size : 42,5 Mb
Release : 2021-10-27
Category : Mathematics
ISBN : 9783030890322

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Topological, Differential and Conformal Geometry of Surfaces by Norbert A'Campo Pdf

This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.

Author : Xianfeng David Gu,Shing-Tung Yau
Publisher : Unknown
Page : 324 pages
File Size : 42,5 Mb
Release : 2008
Category : CD-ROMs
ISBN : UOM:39015080827697

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Computational Conformal Geometry by Xianfeng David Gu,Shing-Tung Yau Pdf

Author : Rafal Ablamowicz,Garret Sobczyk
Publisher : Springer Science & Business Media
Page : 244 pages
File Size : 47,7 Mb
Release : 2003-11-06
Category : Mathematics
ISBN : 0817632573

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Lectures on Clifford (Geometric) Algebras and Applications by Rafal Ablamowicz,Garret Sobczyk Pdf

The subject of Clifford (geometric) algebras offers a unified algebraic framework for the direct expression of the geometric concepts in algebra, geometry, and physics. This bird's-eye view of the discipline is presented by six of the world's leading experts in the field; it features an introductory chapter on Clifford algebras, followed by extensive explorations of their applications to physics, computer science, and differential geometry. The book is ideal for graduate students in mathematics, physics, and computer science; it is appropriate both for newcomers who have little prior knowledge of the field and professionals who wish to keep abreast of the latest applications.