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Author: Liviu I. Nicolaescu
Publisher: World Scientific
ISBN: 9812708537
Category : Mathematics
Languages : en
Pages : 606
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Book Description
The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology.The book's guiding philosophy is, in the words of Newton, that ?in learning the sciences examples are of more use than precepts?. We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue.While we present most of the local aspects of classical differential geometry, the book has a ?global and analytical bias?. We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincar duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-;Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades. Thus, the last part of the book discusses elliptic equations, including elliptic Lpand Hlder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators.The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight.
Author: Liviu I. Nicolaescu
Publisher: World Scientific
ISBN: 9812708537
Category : Mathematics
Languages : en
Pages : 606
Get Book Here
Book Description
The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology.The book's guiding philosophy is, in the words of Newton, that ?in learning the sciences examples are of more use than precepts?. We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue.While we present most of the local aspects of classical differential geometry, the book has a ?global and analytical bias?. We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincar duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-;Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades. Thus, the last part of the book discusses elliptic equations, including elliptic Lpand Hlder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators.The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight.
Author: Izu Vaisman
Publisher: Birkhäuser
ISBN: 3034884958
Category : Mathematics
Languages : en
Pages : 210
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Book Description
This book is addressed to graduate students and researchers in the fields of mathematics and physics who are interested in mathematical and theoretical physics, differential geometry, mechanics, quantization theories and quantum physics, quantum groups etc., and who are familiar with differentiable and symplectic manifolds. The aim of the book is to provide the reader with a monograph that enables him to study systematically basic and advanced material on the recently developed theory of Poisson manifolds, and that also offers ready access to bibliographical references for the continuation of his study. Until now, most of this material was dispersed in research papers published in many journals and languages. The main subjects treated are the Schouten-Nijenhuis bracket; the generalized Frobenius theorem; the basics of Poisson manifolds; Poisson calculus and cohomology; quantization; Poisson morphisms and reduction; realizations of Poisson manifolds by symplectic manifolds and by symplectic groupoids and Poisson-Lie groups. The book unifies terminology and notation. It also reports on some original developments stemming from the author's work, including new results on Poisson cohomology and geometric quantization, cofoliations and biinvariant Poisson structures on Lie groups.
Author: Ana Cannas da Silva
Publisher: Springer
ISBN: 354045330X
Category : Mathematics
Languages : en
Pages : 240
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Book Description
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
Author: Riccardo Benedetti
Publisher: Springer Science & Business Media
ISBN: 3642581587
Category : Mathematics
Languages : en
Pages : 343
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Book Description
Focussing on the geometry of hyperbolic manifolds, the aim here is to provide an exposition of some fundamental results, while being as self-contained, complete, detailed and unified as possible. Following some classical material on the hyperbolic space and the Teichmüller space, the book centers on the two fundamental results: Mostow's rigidity theorem (including a complete proof, following Gromov and Thurston) and Margulis' lemma. These then form the basis for studying Chabauty and geometric topology; a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory; and much space is devoted to the 3D case: a complete and elementary proof of the hyperbolic surgery theorem, based on the representation of three manifolds as glued ideal tetrahedra.
Author: Michael Spivak
Publisher: Westview Press
ISBN: 9780805390216
Category : Science
Languages : en
Pages : 164
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Book Description
This book uses elementary versions of modern methods found in sophisticated mathematics to discuss portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level.
Author: Andrei Moroianu
Publisher: Cambridge University Press
ISBN: 1139463004
Category : Mathematics
Languages : en
Pages : 4
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Book Description
Kähler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi–Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.
Author: Loring W. Tu
Publisher: Springer Science & Business Media
ISBN: 1441974008
Category : Mathematics
Languages : en
Pages : 426
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Book Description
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'.
Author: Spiro Karigiannis
Publisher: Springer Nature
ISBN: 1071605771
Category : Mathematics
Languages : en
Pages : 392
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Book Description
This book, one of the first on G2 manifolds in decades, collects introductory lectures and survey articles largely based on talks given at a workshop held at the Fields Institute in August 2017, as part of the major thematic program on geometric analysis. It provides an accessible introduction to various aspects of the geometry of G2 manifolds, including the construction of examples, as well as the intimate relations with calibrated geometry, Yang-Mills gauge theory, and geometric flows. It also features the inclusion of a survey on the new topological and analytic invariants of G2 manifolds that have been recently discovered. The first half of the book, consisting of several introductory lectures, is aimed at experienced graduate students or early career researchers in geometry and topology who wish to familiarize themselves with this burgeoning field. The second half, consisting of numerous survey articles, is intended to be useful to both beginners and experts in the field.
Author: Werner Ballmann
Publisher: European Mathematical Society
ISBN: 9783037190258
Category : Mathematics
Languages : en
Pages : 190
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Book Description
These notes are based on lectures the author gave at the University of Bonn and the Erwin Schrodinger Institute in Vienna. The aim is to give a thorough introduction to the theory of Kahler manifolds with special emphasis on the differential geometric side of Kahler geometry. The exposition starts with a short discussion of complex manifolds and holomorphic vector bundles and a detailed account of the basic differential geometric properties of Kahler manifolds. The more advanced topics are the cohomology of Kahler manifolds, Calabi conjecture, Gromov's Kahler hyperbolic spaces, and the Kodaira embedding theorem. Some familiarity with global analysis and partial differential equations is assumed, in particular in the part on the Calabi conjecture. There are appendices on Chern-Weil theory, symmetric spaces, and $L^2$-cohomology.
Author: Patrick Eberlein
Publisher: University of Chicago Press
ISBN: 9780226181981
Category : Mathematics
Languages : en
Pages : 460
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Book Description
Starting from the foundations, the author presents an almost entirely self-contained treatment of differentiable spaces of nonpositive curvature, focusing on the symmetric spaces in which every geodesic lies in a flat Euclidean space of dimension at least two. The book builds to a discussion of the Mostow Rigidity Theorem and its generalizations, and concludes by exploring the relationship in nonpositively curved spaces between geometric and algebraic properties of the fundamental group. This introduction to the geometry of symmetric spaces of non-compact type will serve as an excellent guide for graduate students new to the material, and will also be a useful reference text for mathematicians already familiar with the subject.
