Algebraic groups and Lie groups are important in most major areas of mathematics, occuring in diverse roles such as the symmetries of differential equations and as central figures in the Langlands program for number theory. In this book, Professor Borel looks at the development of the theory of Lie groups and algebraic groups, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. As the starting point of this passage from local to global, the author takes Lie's theory of local analytic transformation groups and Lie algebras. He then follows the globalization of the process in its two most important frameworks: (transcendental) differential geometry and algebraic geometry. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter.The essays in the first part of the book survey various proofs of the full reducibility of linear representations of $SL 2M$, the contributions H. Weyl to representation and invariant theory for Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large. The second part of the book starts with Chapter V describing the development of the theory of linear algebraic groups in the 19th century. Many of the main contributions here are due to E. Study, E. Cartan, and above all, to L. Maurer. After being abandoned for nearly 50 years, the theory was revived by Chevalley and Kolchin and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on various aspects of the works of Chevalley on Lie groups and algebraic groups and Kolchin on algebraic groups and the Galois theory of differential fields. The author brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep appreciation of the underlying mathematics. His lifelong involvement and his historical research in the subject give him a special appreciation of the story of its development.

This is an introduction to geometrical topics that are useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing readers for the study of modern treatments of mechanics, gauge fields theories, relativity and gravitation. The order of presentation corresponds to that used for the relevant material in theoretical physics: the geometry of affine spaces, which is appropriate to special relativity theory, as well as to Newtonian mechanics, is developed in the first half of the book, and the geometry of manifolds, which is needed for general relativity and gauge field theory, in the second half. Analysis is included not for its own sake, but only where it illuminates geometrical ideas. The style is informal and clear yet rigorous; each chapter ends with a summary of important concepts and results. In addition there are over 650 exercises, making this a book which is valuable as a text for advanced undergraduate and postgraduate students.

Book 3 in the Princeton Mathematical Series. Originally published in 1950. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.

An examination of approaches to easy-to-understand but difficult-to-solve mathematical problems, this classic text begins with a discussion of Dirichlet's principle and the boundary value problem of potential theory, then proceeds to examinations of conformal mapping on parallel-slit domains and Plateau's problem. Also explores minimal surfaces with free boundaries and unstable minimal surfaces. 1950 edition.

This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hoermander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincare's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schroedinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects.

Mathematical No/ex, 27

Originally published in 1981.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

An introductory textbook on the differential geometry of curves and surfaces in three-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. With problems at the end of each section, and solutions listed at the end of the book. 99 illustrations.

This text is an invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behaviour of soap films. When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This tool makes possible new analyses and new results Plateau's problem and varifolds live in the world of geometric measure theory, where differential geometry and measure theory combine to solve problems which have variational aspects. This book aims to encourage young mathematicians to study this field.

For a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin (or $\textrm {spin}^\mathbb{C}$) structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants.In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm {spin}^\mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature.Considerations of Killing spinors and solutions of the twistor equation on $M$ lead to results about whether $M$ is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag.

The Teichmuller space $T(X)$ is the space of marked conformal structures on a given quasi conformal surface $X$. This volume uses quasi conformal mapping to give a unified and up-to-date treatment of $T(X)$. Emphasis is placed on parts of the theory applicable to non compact surfaces and to surfaces possibly of infinite analytic type. The book provides a treatment of deformations of complex structures on infinite Riemann surfaces and gives background for further research in many areas. These include applications to fractal geometry, to three-dimensional manifolds through its relationship to Kleinian groups, and to one-dimensional dynamics through its relationship to quasi symmetric mappings. Many research problems in the application of function theory to geometry and dynamics are suggested.

This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. The approach is simple: a mapping is called smooth if it maps smooth curves to smooth curves. Up to Frechet spaces, this notion of smoothness coincides with all known reasonable concepts. In the same spirit, calculus of holomorphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups.Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations.

This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Bohm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

Zu Recht wird Albert Einsteins Entdeckung der Allgemeinen Relativitatstheorie bewundert, denn ihre Erkenntnisse haben unseren Blick auf das Universum grundlegend verandert. Aus mathematischer Perspektive basiert die Theorie auf zentralen Aussagen der Riemann'schen Geometrie. Dieses Buch liefert eine didaktisch aufbereitete und interdisziplinare Einfuhrung in die Geometrie der Allgemeinen Relativitatstheorie. Ausgehend von Einsteins typischen UEberlegungen und Gedankenexperimenten werden die Prinzipien der Relativitatstheorie erarbeitet und mit den zugrundeliegenden mathematischen Konzepten der Differentialgeometrie verknupft. Der Autor bietet durch die Verbindung beider Fachdisziplinen sowohl fur Studierende der Physik als auch der Mathematik die Moeglichkeit, in eine der faszinierendsten Theorien der Physik einzutauchen.

This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists.

One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.

This book introduces the tools of modern differential geometry--exterior calculus, manifolds, vector bundles, connections--and covers both classical surface theory, the modern theory of connections, and curvature. Also included is a chapter on applications to theoretical physics. The author uses the powerful and concise calculus of differential forms throughout. Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. Nearly 200 exercises make the book ideal for both classroom use and self-study for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering.

Nigel Hitchin is one of the world's foremost figures in the fields of differential and algebraic geometry and their relations with mathematical physics, and he has been Savilian Professor of Geometry at Oxford since 1997. Geometry and Physics: A Festschrift in honour of Nigel Hitchin contain the proceedings of the conferences held in September 2016 in Aarhus, Oxford, and Madrid to mark Nigel Hitchin's 70th birthday, and to honour his far-reaching contributions to geometry and mathematical physics. These texts contain 29 articles by contributors to the conference and other distinguished mathematicians working in related areas, including three Fields Medallists. The articles cover a broad range of topics in differential, algebraic and symplectic geometry, and also in mathematical physics. These volumes will be of interest to researchers and graduate students in geometry and mathematical physics.

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic.

Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

This book treats the two-dimensional non-linear supersymmetric sigma model or spinning string from the perspective of supergeometry. The objective is to understand its symmetries as geometric properties of super Riemann surfaces, which are particular complex super manifolds of dimension 1|1. The first part gives an introduction to the super differential geometry of families of super manifolds. Appropriate generalizations of principal bundles, smooth families of complex manifolds and integration theory are developed. The second part studies uniformization, U(1)-structures and connections on Super Riemann surfaces and shows how the latter can be viewed as extensions of Riemann surfaces by a gravitino field. A natural geometric action functional on super Riemann surfaces is shown to reproduce the action functional of the non-linear supersymmetric sigma model using a component field formalism. The conserved currents of this action can be identified as infinitesimal deformations of the super Riemann surface. This is in surprising analogy to the theory of Riemann surfaces and the harmonic action functional on them. This volume is aimed at both theoretical physicists interested in a careful treatment of the subject and mathematicians who want to become acquainted with the potential applications of this beautiful theory.

Professor Atiyah is one of the greatest living mathematicians and is renowned in the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still actively involved in the mathematics community. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into seven volumes, with the first five volumes divided thematically and the sixth and seventh arranged by date. This seven volume set of the collected works of Professor Sir Michael Atiyah, includes: Collected Works: Volume 1: Early Papers; General Papers Collected Works: Volume 2: K-Theory Collected Works: Volume 3: Index Theory: 1 Collected Works: Volume 4: Index Theory: 2 Collected Works: Volume 5: Gauge Theories Collected Works: Volume 6: Publications between 1987 and 2002 New for 2014: Collected Works: Volume 7: 2002-2013, including Sir Michael's work on skyrmions; K-theory and cohomology; geometric models of matter; curvature, cones and characteristic numbers; and reflections on the work of Riemann, Einstein and Bott.

Professor Atiyah is one of the greatest living mathematicians and is renowned in the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still actively involved in the mathematics community. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into seven volumes, with the first five volumes divided thematically and the sixth and seventh arranged by date. This seventh volume in Michael Atiyah's Collected Works contains a selection of his publications between 2002 and 2013, including his work on skyrmions; K-theory and cohomology; geometric models of matter; curvature, cones and characteristic numbers; and reflections on the work of Riemann, Einstein and Bott.

Dieses Buch richtet sich an Studierende der Mathematik in der Vertiefungsphase des Bachelor-Studiums. Ausgehend von den Grundvorlesungen Analysis I-III und Lineare Algebra I-II werden zunachst die Grundlagen der Differentialtopologie von Mannigfaltigkeiten behandelt, dann die Grundlagen der Rie-mannschen Geometrie, und anschliessend wird in die Geometrie von homogenen und symmetrischen Raumen eingefuhrt. Das Buch soll einen moeglichst vollstandigen Zugang zur Differentialgeometrie homogener Raume bieten, mit kompletten Beweisen. Es enthalt zahlreiche UEbungsaufgaben, Loesungen und Hinweise zu einigen Aufgaben findet man am Ende des Buches.

Die Elementare Differentialgeometrie (nicht nur) fur Informatiker entstand aus einer Vorlesung an Hochschule fur Angewandte Wissenschaften Hamburg (HAW) uber mathematische Methoden der Computergrafik. Statt wie in der Computergrafik ublich Beispiele zu horten wird eine systematische doch elementare und spannende Geschichte erzahlt, in der man sich sofort festliest. Das Konzept bindet ca. 80 Videos des Autors mit ein sowie zahlreiche Abbildungen und konkrete Programmcodes und UEbungsaufgaben.