Download or read book Cohomology and Differential Forms written by Izu Vaisman and published by Courier Dover Publications. This book was released on 2016-07-28 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Self-contained development of cohomological theory of manifolds with various sheaves and its application to differential geometry covers categories and functions, sheaves and cohomology, fiber and vector bundles, and cohomology classes and differential forms. 1973 edition.
Download or read book Differential Forms in Algebraic Topology written by Raoul Bott and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.
Download or read book Cohomology and Differential Forms written by Izu Vaisman and published by Courier Dover Publications. This book was released on 2016-08-17 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph explores the cohomological theory of manifolds with various sheaves and its application to differential geometry. Based on lectures given by author Izu Vaisman at Romania's University of Iasi, the treatment is suitable for advanced undergraduates and graduate students of mathematics as well as mathematical researchers in differential geometry, global analysis, and topology. A self-contained development of cohomological theory constitutes the central part of the book. Topics include categories and functors, the Čech cohomology with coefficients in sheaves, the theory of fiber bundles, and differentiable, foliated, and complex analytic manifolds. The final chapter covers the theorems of de Rham and Dolbeault-Serre and examines the theorem of Allendoerfer and Eells, with applications of these theorems to characteristic classes and the general theory of harmonic forms.
Download or read book From Calculus to Cohomology written by Ib H. Madsen and published by Cambridge University Press. This book was released on 1997-03-13 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introductory textbook on cohomology and curvature with emphasis on applications.
Download or read book Residues and Traces of Differential Forms via Hochschild Homology written by Joseph Lipman and published by American Mathematical Soc.. This book was released on 1987 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: Requiring only some understanding of homological algebra and commutative ring theory, this book gives those who have encountered Grothendieck residues in geometry or complex analysis an understanding of residues, as well as an appreciation of Hochschild homology.
Download or read book Geometry of Differential Forms written by Shigeyuki Morita and published by American Mathematical Soc.. This book was released on 2001 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the times of Gauss, Riemann, and Poincare, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory. The book can serve as a textbook for undergraduate students and for graduate students in geometry.
Download or read book Differential Geometry written by Loring W. Tu and published by Springer. This book was released on 2017-06-01 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.
Download or read book Rational Homotopy Theory and Differential Forms written by Phillip Griffiths and published by Springer Science & Business Media. This book was released on 2013-10-02 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham’s theorem on simplicial complexes. In addition, Sullivan’s results on computing the rational homotopy type from forms is presented. New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma *Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theory With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.
Download or read book Algebraic Topology Via Differential Geometry written by M. Karoubi and published by Cambridge University Press. This book was released on 1987 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry.
Download or read book Introductory Lectures on Equivariant Cohomology written by Loring W. Tu and published by Princeton University Press. This book was released on 2020-03-03 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into K-theory and algebraic geometry, but it is in algebraic topology that the concepts are the most transparent and the proofs are the simplest. One of the most useful applications of equivariant cohomology is the equivariant localization theorem of Atiyah-Bott and Berline-Vergne, which converts the integral of an equivariant differential form into a finite sum over the fixed point set of the group action, providing a powerful tool for computing integrals over a manifold. Because integrals and symmetries are ubiquitous, equivariant cohomology has found applications in diverse areas of mathematics and physics. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localization theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.
Download or read book Differential Forms on Singular Varieties written by Vincenzo Ancona and published by CRC Press. This book was released on 2005-08-24 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Forms on Singular Varieties: De Rham and Hodge Theory Simplified uses complexes of differential forms to give a complete treatment of the Deligne theory of mixed Hodge structures on the cohomology of singular spaces. This book features an approach that employs recursive arguments on dimension and does not introduce spaces of hig
Download or read book Connections Curvature and Cohomology V1 written by and published by Academic Press. This book was released on 1972-07-31 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: Connections, Curvature, and Cohomology V1
Download or read book An Introduction to Manifolds written by Loring W. Tu and published by Springer Science & Business Media. This book was released on 2010-10-05 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.
Download or read book Cohomology Operations and Applications in Homotopy Theory written by Robert E. Mosher and published by Courier Corporation. This book was released on 2008-01-01 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: Cohomology operations are at the center of a major area of activity in algebraic topology. This treatment explores the single most important variety of operations, the Steenrod squares. It constructs these operations, proves their major properties, and provides numerous applications, including several different techniques of homotopy theory useful for computation. 1968 edition.
Download or read book Differential Forms written by Guillemin Victor and published by World Scientific. This book was released on 2019-03-20 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: There already exist a number of excellent graduate textbooks on the theory of differential forms as well as a handful of very good undergraduate textbooks on multivariable calculus in which this subject is briefly touched upon but not elaborated on enough.The goal of this textbook is to be readable and usable for undergraduates. It is entirely devoted to the subject of differential forms and explores a lot of its important ramifications.In particular, our book provides a detailed and lucid account of a fundamental result in the theory of differential forms which is, as a rule, not touched upon in undergraduate texts: the isomorphism between the Čech cohomology groups of a differential manifold and its de Rham cohomology groups.
Download or read book Differential Forms written by Steven H. Weintraub and published by Academic Press. This book was released on 1997 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is one of the first to treat vector calculus using differential forms in place of vector fields and other outdated techniques. Geared towards students taking courses in multivariable calculus, this innovative book aims to make the subject more readily understandable. Differential forms unify and simplify the subject of multivariable calculus, and students who learn the subject as it is presented in this book should come away with a better conceptual understanding of it than those who learn using conventional methods. * Treats vector calculus using differential forms * Presents a very concrete introduction to differential forms * Develops Stokess theorem in an easily understandable way * Gives well-supported, carefully stated, and thoroughly explained definitions and theorems. * Provides glimpses of further topics to entice the interested student
Download or read book Foundations of Differentiable Manifolds and Lie Groups written by Frank W. Warner and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.
